PERSONAL NOTES IN OPERATOR ALGEBRAS AND OPERATOR
THEORY
MINH KHA
Abstract. This note collects some facts, theorems in operator algebras and operator theory for fullfilling my understanding in my studying progress. Part I contains
some fundamental notes which I collect from many textbooks and surveys. I will
either try to provide some unclear details or solve some basic but important exercises. Other parts might include some parts in published articles which I try to fill
in more details. The choices of the topics here are due to my personal taste. Notes
are very incomplete and contains certain errors. Comments are welcome.
1. Some fundamental facts in Operator Theory and Operator
Algebras
1.1. Some preliminaries in spectral theory of bounded linear operators.
We start with a simple statement about the relation of the support of the spectral
measure and the spectrum of a normal operator T .
Proposition 1.1. Let T ∈ B(H) be a normal operator. Then a complex number
λ ∈ σ(T ) if and only if ET (U ) 6= 0 for every open subset U containing λ.
Here is an important statement concerning isolated points in spectrum of normal
operators (Exercise 11, p.75 of [10]):
Proposition 1.2. Let T be a normal operator in B(H) where H is a Hilbert space.
Then if x is an isolated point of the spectrum σ(T ), we have x is an isolated eigenvalue
of T and the spectral measure ET ({x}) is the projection onto the eigenspace Ker(T −
xI). Moreover, T has a nontrivial invariant closed vector subspace if dim(H) > 1.
Proof. The function χ{x} (z − x) is identical to the zero function on σ(T ). Hence (T −
xI)ET ({x}) = 0 by continuous functional calculus and so ET ({x}) is a subprojection
of Ker(T − xId). Since the subspace Ker(T − x) is an invariant subspace under T
and T ∗ , the projection P onto Ker(T − x) should be regarded as a characteristic
function χE in L∞ (σ(T ), µ) for some E ∈ Bσ(T ) and so x ∈ E. Since (T − x)P = 0,
χE (z)(z − x) = 0, µ − a.e z ∈ σ(T ) or µ{E \ {x}} = 0. This implies that χE = χ{x}
or P = ET ({x}).
Because x is isolated in σ(T ), it follows that ET ({x}) > 0 and this means the
eigenspace Ker(T −x) is nontrivial or x is an eigenvalue for T . T ET ({x}) = xET ({x})
yields that the range of ET ({x}) is a nontrivial invariant subspace for T in the case
ET ({x}) < 1. Otherwise, T = xId and this case is totally trivial if dim(H) > 1. 2000 Mathematics Subject Classification. 20C07, (20E99).
Key words and phrases. .
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MINH KHA
From the above proof, for any T normal and any λ ∈ σ(T ), we always have
ET ({x}) = Ker(T − λ). In particular, ET ({0}) = Ker(T ), ET ∗ ({0})⊥ = Ran(T ).
Proposition 1.3. The approximate point spectrum
σap (T ) = {λ ∈ C : T − λ is not bounded below}
(1)
is a closed subset which contains the boundary of the spectrum ∂σ(T ). If T is normal
then σ(T ) = σap (T ).
Since we are interested in describing the structure of the set of all eigenvalues of
a bounded linear operator, the following statement plays a special role in studying
quasisimilarity in B(H):
Theorem 1.4. The point spectrum σp (T ) is equal to the union of all holes or pseudoholes of σe (T ) whose index belongs to N ∪ {+∞} together with isolated points of
σ(T ) \ σe (T ) and the holes H of σe (T ) such that H ⊂ σ(T ) and its index is 0.
1.2. Review on Fredholm operators and Index theory, The Essential Spectrum, The Toeplitz Extension, Essential Normal Operators.
Definition 1.5. Let T ∈ B(H, K) where H, K are two Hilbert spaces. T is called a
Fredholm operator iff Ker(T ) and Coker(T ) are finite dimensional vector subspaces
(i.e the quotient vector space K/Ran(T ) is finite dimensional). Equivalently, T is
Fredholm if and only if T has closed range and Ker(T ) and Ker(T ∗ ) are finite
dimensional.Then Index(T ) = dim(Ker(T ))−dim(Coker(T )) is the integer quantity,
i.e Fredholm index of a Fredholm operator T .
Remark 1.6. The range of T is always closed. Indeed, this is true if the cokernel of T
is finite dimensional as the following argument shows. First, we take the restriction
of T onto the orthogonal complement of the kernel of T to get an injective bounded
linear operator. Since the image of this map is co-finite dimensional, it is easy to
extend it to a bijective bounded linear map T 0 : Ker(T )⊥ ⊕ Cn → K. Hence,
T 0 is a homeomorphism and since Ker(T )⊥ is closed in Ker(T )⊥ ⊕ Cn , Ran(T ) =
T 0 (Ker(T )⊥ ) is closed as we wanted.
Remark 1.7. Let U, V be Fredholm operators. Then U V is Fredholm with ind(U V ) =
ind(U ) + ind(V ).
Examples: (i) If H, K are finite dimensional, then every operator from H to K is
Fredholm with index dim(H) − dim(K).
(ii) If T is invertible, then T is Fredholm with index zero.
(iii) Let S be the unilateral shift. Then S is Fredholm with index -1. Hence, S n is
Fredholm with index −n and (S ∗ )n is Fredholm with index n.
Definition 1.8. σe (T ), σle (T ), σre (T ) are the essential, left essential, right essential
spectrum of T ∈ B(H), i.e the set of all scalars λ such that π(T − λ) is not invertible,
left invertible, right invertible in the Calkin algebra, where π : B(H) → B(H)/K(H)
is the canonical map onto the Calkin algebra.
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Proposition 1.9. For each T ∈ B(H), σe (T ), σle (T ), σre (T ) are non-empty compact
T subsets of σ(T ). Moreover, σe (T ) = {λ ∈ C : T − λ is not Fredholm} ⊂
K∈K(H) σ(T + K) = σ(T ) \ {λ ∈ C : T − λ is Fredholm with index zero}, and
also
T ∂σe (T ) ⊂ σle (T ) ∩ σre (T ). Hence, if C \ σe (T ) is connected, then σe (T ) =
K∈K(H) σ(T + K).
Examples: The essential spectrum of the unilateral shift S is the whole circle T.
Note that σ(T ) \ σe (T ) is called the discrete spectrum σdiscr (T ). The essential
spectrum radius re (T ) = max{|λ| : λ ∈ σe (T )} is equal to the spectrum radius r(T ).
1.3. Ideals and Hereditary C*-subalgebras. A notable fact on hereditary C*subalgebra is the following one
Theorem 1.10. Let B be a separable hereditary C*-subalgebra in A then there exists
a positive element a ∈ B such that B = Her(a) = C ∗ (aAa).
P
−n
un where
Proof. Since B is separable, B is σ-unital. So we just let a = ∞
n=1 2
{un } is a sequence of approximate units of B.
Counterexample: For non-separable Hilbert space H, the two sided ideal K(H)
is not generated by a single positive element u as a hereditary-C*-subalgebra of
B(H). Indeed, suppose for contradiction that K(H) = Her(u) then using rank one
projections x ⊗ x (x ∈ H) , H is the closure of the range of u. Since u is a compact
operator, its range must be separable. Thus H is separable (contradiction!).
However, for von Neumann algebra M , we still have a similar conclusion for the
class of weak (strong) operator closed hereditary C*-subalgebra A in M , i.e there
is (unique) a projection p ∈ A such that A = pM p. For weak operator closed left
(two-sided) ideal in M , A = pM (respectively p ∈ Z(M )).
For unital case, we could slightly change the definition of hereditary C*-subalgebra
by an approximate version.
Theorem 1.11. If B is a hereditary C*-subalgebra of a unital C*-algebra A and
a ∈ A+ . If for any > 0, there exists b ∈ B + such that a ≤ b + then a ∈ B.
Proof. By assumption, for each > 0, we find one b ∈ B + such that a ≤ (b + )2
or (b + )−1 a(b + )−1 ≤ 1. Recall that this is in a similar situation while working
with b (b + )−1 as an approximate unit for a C*-algebra: We easily get a1/2 =
lim→0 a1/2 b (b + )−1 . Thus a = lim→0 (b + )−1 b ab (b + )−1 . Since b (b + )−1 ≤
b −1 , it follows that b (b + )−1 ∈ B. Now everthing follows from the previous
theorem.
Theorem 1.12. Let J be a closed ideal in a hereditary C*-subalgebra B of a C*algebra A. Then we can find another closed ideal I in A such that J = I ∩ B.
Proof. We sketch the proof. Let I = AJA. It’s easy to check I is a closed ideal in A.
Since B is hereditary, B ∩ I = BIB. Then the following relations are easy to check:
B ∩ I = (BAJ)J(JAB) ⊂ BJB = J.
An important corrolary is the following fact
Corollary 1.13. Any hereditary C*-subalgebra of a simple C*-algebra is also simple.
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MINH KHA
Proof. Straightforward application of the previous theorem.
Of course, this corollary is wrong if we omit the hereditary condition.
It’s well-known that a C*-subalgebra of a nuclear C*-algebra is not always nuclear.
Fortunately, we have the following result:
Theorem 1.14. A hereditary subalgebra B of a nuclear C*-algebra A is again nuclear.
In particular, nuclearity passes to closed ideals.
Proof. Let {en } be an approximate unit for B then the c.c.p maps θn : A → B
defined by θn (a) = en aen have the properties that kθn (b) − bk → 0 for b ∈ B. Note
that the restriction idA |B : B → A is still a nuclear map. Then the sequence of c.c.p
maps φn = idA |B ◦ θn satisfies that φn |B → idB in the point-norm topology. Thus
idA |B : B → B is a nuclear map by Exercise 2.1.8 in [3].
For completeness, I will type up the proof of this important exercise: Let θ : A → B
be a nuclear map and assume C is a C*-subalgebra of B such that θ(A) ⊂ C and
there exist c.c.p maps θn : B → C such that kθn (c) − ck → 0 for every c ∈ C. Then
θ0 : A → C is nuclear. Hence, it is particularly true in the case that there is a
conditional expectation Φ : B → C.
Indeed, we first find c.c.p maps ψn : A → Mk(n) (C) and φn : Mk(n) (C) → B such
that φn ◦ ψn → θ in point-norm topology. Let Φn = θn ◦ φn : Mkn (C) → C then
Φn ◦ ψn → θ0 by assumption, which yields the conclusion.
A simple corollary is that any approximate homogeneous (AH) C*-algebra is nuclear. From the above proof, a very useful picture for hereditary subalgebras is that
there exists a sequence (or net for non-separable case) of c.c.p maps that approximates in the point-norm topology to “the conditional expectation from the original
C* algebra onto the hereditary one”. Of course, it is not neccessary to have a conditional expectation onto a hereditary C*-subalgebra or even a closed two sided ideal
as the following elementary proposition shows
Proposition 1.15. There exists no conditional expectation from B(H) onto K(H)
for infinite dimensional Hilbert space H.
Proof. Suppose for contradiction. Let E be that conditional expectation. To get a
contradiction, it suffices to analyze E(1) = x. Note that x is a positive compact
operator. Let y be any compact operator. Then E(y ∗ y) = y ∗ xy and thus, h(1 −
x)y(ξ), y(ξ)i = 0 for any ξ ∈ H. Since the action of compact operators on H is
transitive, the numerical range of 1 − x is 0. Since σ(1 − x) ⊆ W (1 − x), we conclude
that 1 = x since x is self-adjoint. This is obviously a contradiction.
Definition 1.16. A well-supported element x in a C*-algebra A satisfies
1.4. Polar decomposition in C*-algebras, General Comparision theory. To
begin, we start with the von Neumann algebra case first. As we know, e.g [?Takesaki1],
any element T in a von Neumann algebra M has a polar decomposition T = u|T | =
(T T ∗ )1/2 u, where u ∈ M is a partial isometry and Ker(u) = Ker(T ) = Ker(|T |) or
another equivalent condition is that the initial space of U is r(|T |) and the final space
of U is r(T ). The polar decomposition is unique in the sense that for such any decomposition T = W H where H ∈ M + , W ∈ M is a partial isometry such that r(H) is the
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initial space of W then H = |T |, W = u. In particular, if ker(T ) = ker(T ∗ ) = {0}
then u is a unitary. However, if the von Neumann algebra M is finite, we have the
following thing:
Lemma 1.17. If T ∈ M , where M is finite then there exists a unitary U ∈ M such
that T = U |T |
Proof. From the polar decomposition of T ∈ M , we have a partial isometry u ∈ M
such that T = u|T |. Let p = r(|T |) so u = up. Denote U = u + 1 − p ∈ M then
U U ∗ = (u + 1 − p)(u∗ + 1 − p) = p + 1 − p + (1 − p)u∗ + u(1 − p) = 1. By finiteness,
U is a unitary in M . Also, T = u|T | + (1 − p)|T | = U |T |.
Note that all above statements can be applied to the case when T is a densely
defined unbounded operator which is affiliated to M .
1.5. Simplicity, Separability related to K(H),B(H).
Theorem 1.18. If H is an infinite dimensional Hilbert space, then K(H) , the Calkin
algebra Q(H) = B(H)/K(H) are simple C*-algebras.
Proposition 1.19. For separable Hilbert space H, K(H) is a separable C*-algebra
while B(H) or Q(H) is non-separable.
Proof. Since K(H) is the closure span of the rank one operator en ⊗ em , where {en } is
an orthornormal basis of H, we see immediately the first fact. For the later fact, since
we can always find an uncountable family of (infinite range) projections pS where S
is a (infinite) subset of {en }, the non-separability of B(H) and Q(H) come from the
fact that kpS − pS 0 k = 1 whenever S 6= S 0 .
Again, this shows why SOT or WOT are the correct topologies to deal with von
Neumann algebra. In the case of a separable Hilbert space H, the closed unit ball of
B(H) is always separable under the SOT-topology and even that it is also metrizable
(also under the WOT-topology). Note that the latter topology is straightforward to
check since the closed unit ball of B(H) is weak operator compact and hence it is
separable since a compact metric space is separable. For completeness, I will present
the proof of separability of the closed unit ball of B(H) in SOT. Actually, this is
one of many equivalent characterization of a von Neumann algebra with separable
predual.
Proposition 1.20. The closed unit ball S of B(H) is separable under the strong
operator topology if H is a separable Hilbert space.
Proof. We see easily that K(H) is strongly dense in B(H) and from the previous fact
we know that K(H) is norm-separable. By Kaplansky density theorem, the closed
unit ball of K(H) is strongly dense in S and hence it proves all.
1.6. Some useful facts on Nuclearity and Exactness in C*-algebras. First,
this subsection is merely a result of gathering very basic and small facts related to
nuclearity and exactness which . . . I cannot find in any textbook with full details
(since they are so fundamental probably!). However, it should not be surprised that
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MINH KHA
I use exercises in Chapter II of [3] for serving this subsection. Hence, I do not recall
definitions as well as other properties which are showed and appeared in their book.
I do recall that nuclearity is equivalent to local nuclearity. This means, a C*algebra A is nuclear iff for any finite subset F ⊂ A and > 0, there exists a nuclear
C*-subalgebra B of A such that maxx∈F dist(x, B) < (The proof of this is just the
Arveson’s extension theorem and several lines of applying triangle inequalities). A
very important corollary is that the inductive limit with injective connecting maps
of nuclear C*-algebras is still nuclear. Hence, any AF -algebra is nuclear. If we use
the deep fact that any quotient of nuclear C*-algebra is nuclear, we get the case of
full generality for inductive limits of nuclear (or exact) C*-algebras. Similarly, the
externally local exactness implies exactness. Before giving a proof, we need a useful
preliminary fact
Proposition 1.21. (Independent of representation) Assume that A ⊂ B(H) is a
concretely represented exact C*-algebra. Then there exist c.c.p maps ϕn : A →
Mk(n) (C) and φn : Mk(n) (C) → B(H) such that ka − φn ◦ ϕn (a)k → 0 for all a ∈ A.
Proposition 1.22. (Externally Local Exactness) Assume A ⊂ B(H) is a concretely
represented C*-algebra such that for any finite subset F ⊂ A and > 0 there exists
an exact C*-subalgebra B of B(H) such that maxx∈F dist(x, B) < . Then A is also
exact.
For non-separable case, probably we do not need to be worried too much when
dealing with exactness and nuclearity by the following proposition (which is Exercise
2.3.8 in [3])
Proposition 1.23. A is exact iff every separable C*-subalgebra of A is exact. For the
nuclear case, A is nuclear iff for every separable C*-subalgebra B of A, there exists a
nuclear separable C*-subalgebra C of A such that B ⊂ C.
1.7. C*-norm on tensor product of C*-algebras. Any C*-norm on A ⊗alg B
lies between k.kmax and k.kmin , and all of them are cross-norms. The minimal tensor
product norm is the smallest C*-norm on A ⊗alg B. This is the Takesaki’s theorem
whose arguments are beautiful soft analysis.
Thus given any k.kα -C*-norm on A ⊗alg B, there are surjective ?-homomorphisms
A ⊗max B → A ⊗α B → A ⊗min B. Hence, an injective ?-homomorphism on A ⊗alg B
which can be extended on A ⊗min B would be injective on all A ⊗min B, while this is
not true for A ⊗max B.
It is easy to verify that if A ⊗min B is simple then A, B are two simple C*-algebras.
Conversely, this is also true. More generally, we have the following theorem
Theorem 1.24. A ⊗α B is simple iff A, B are simple and k.kα = k.kmin .
Proof. We only need to prove the converse. Assume A, B are simple and k.kα =
k.kmin . To prove A ⊗α B is simple, it suffices to show that every irreducible representation of A ⊗α B is faithful.
1.8. Notes on tensor products of B(H) ⊗ B(K). Given Hilbert spaces H, K, we
want to examine in this subsection properties of different tensor products of B(H) and
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B(K). Consider the finite dimensional case first, assume dim(H) = m, dim(K) = n
where m, n ∈ N. The following simple proposition describes the algebraic tensor
product of these two matrix algebras Mm , Mn :
Proposition 1.25. Mm (C) ⊗ Mn (C) ' Mmn (C) and thus, U(m) ⊗ U(n) ' U(mn)
as metric groups.
Proof. We sketch the proof here. Given U ∈ Mm , V ∈ Mn then we define U ∗V ∈ Mmn
by setting its (mk+r+1, ml+s+1) entry to be Ur+1,s+1 Vk+1,l+1 where 0 ≤ k, l ≤ n−1
and 0 ≤ r, s ≤ m − 1. Then it is not hard to check that (U U 0 ) ∗ (V V 0 ) = (U ∗ V )(U 0 ∗
V 0 ), ∀U, U 0 ∈ Mm , V, V 0 ∈ Mn . Hence the map U ⊗ V → U ∗ V is a well-defined and
a *-homomorphism from Mm (C) ⊗ Mn (C) to Mmn (C). It is clear that this map is
trace-preserving and hence, it is an *-isometric mapping with respect to the HilbertSchmidt norms k.k2 . Since their dimensions are equal to mn, we conclude they
are *-isomorphic (As von Neumann algebras). When we restrict this isomorphism
to U(m) ⊗ U(n), we get a group *-isomorphism since the image is exactly U(mn).
Indeed, take any A ∈ U(mn) then we can find U, V ∈ Mm , Mn such that A = U ∗ V
and hence U ∗ U ∗ V ∗ V = U U ∗ ∗ V V ∗ = 1. This implies U U ∗ = αI, V V ∗ = α−1 I for
some α > 0 and vice versa. Then let U 0 = α−1/2 U, V 0 = α1/2 V , A = U 0 ∗ V 0 while
U 0 , V 0 are unitaries.
Note that the above isomorphism is nothing else than AdU where U : Cm ⊗ Cn →
C is the canonical shuffling. This is also called the Kronecker product map of V and
U . Interestingly, the Schur product map of two matrices A, B ∈ Mn is V ∗ (A ⊗ B)V
where V : Cn → Cn ⊗ Cn is the isometry which sends each vector ei to ei ⊗ ei .
Finally, the above proof works for replacing C by a C*-algebra A. In particular, by a
permutation matrix, Mm (Mn (A)) ' Mn (Mm (A)). Details are in Chapter 3,8 of [17].
When passing to infinite dimensional cases, this is no longer true if we consider
the algebraic or minimal tensor product as C*-algebras. However, we always have
mn
Proposition 1.26. Assume H, K are Hilbert spaces whose dimensions could be infinity. Then the canonical map from the algebraic tensor product of B(H) and B(K)
into B(H ⊗ K) is always injective. However, it is not true that we always have
B(H) ⊗min B(K) ' B(H ⊗ K) (e.g H = K = `2 ) and this is true if and only if
one of H or K is finite dimensional. Moreover, they are *-isomorphic on the von
Neumann algebra level, i.e B(H)⊗B(K) ' B(H ⊗ K).
1.9. Normal morphisms and Ideals. Here are some fundamental results related
to normal homomorphism between von Neumann algebras. The reference is section
2.8 of [15].
Proposition 1.27. Let φ be an isomorphism between two von Neumann algebras M
and N . Then φ is normal.
Theorem 1.28. Let φ be a normal morphism between two von Neumann algebra A
and B. Then kerφ and φ(A) are strong closed in A and B respectively.
Proposition 1.29. Let M be a von Neumann algebra. For each strongly closed
hereditary C*-subalgebra N of M , there is a unique projection p ∈ M such that
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MINH KHA
N = pM p. For each strong closed left ideal N of M , there is a unique projection
p ∈ M such that N = M p. If N is a strongly closed ideal, then there is a unique
central projection p ∈ Z(M ) such that N = M p. In all these cases, p is the unit of
the von Neumann subalgebra N .
Here is an important corollary which characterizes the structure of the range of a
normal morphism:
Corollary 1.30. The image of a von Neumann algebra M under a normal morphism
is isomorphic to M q where q ∈ Z(M ) is a central projection.
Proof. The kernel of a normal morphism is M p for some central projection
L p by the
previous theorem and proposition. Thus, if q = 1 − p then M = M p M q. Then
it is clear that the image of M under a normal morphism is isomorphic to M q since
M/M p is isomorphic to M q.
It’s clear that a positive and normal map between two von Neumann algebras is
also strongly continuous. However, there are examples of normal and positive maps
that are not strong-strong continuous. Anyway, the following theorem describes the
connection between normality and SOT on norm-bounded subsets of a u.c.p map:
Theorem 1.31. Let φ : M → N be a normal u.c.p map between two von Neumann
algebras. Then φ is SOT-continuous on norm-bounded sets. Thus, in particular,
any normal, unital *-homomorphism between two von Neumann algebras is strongly
continuous on norm-bounded sets.
Corollary 1.32. Suppose that φ : M → N is an isomorphism. Then the restriction
of φ : M1 → N1 is a homeomorphism when we equip SOT or WOT topologies on
those unit balls of M, N .
1.10. Noncommutative Lusin theorem and Up-Down theorem. Here is the
noncommutative version of Lusin theorem
Theorem 1.33. Let A be a C*-algebra acting nondegenerately on a Hilbert space
H. Given any > 0, y ∈ A00sa and projection p0 ∈ A00 and a finite subset of vectors
F ⊂ H, there exists x ∈ Asa and a subprojection p ∈ A00 of p0 which is −within
p0 on F (i.e kp(h) − p0 (h)k < , ∀h ∈ F) such that kxk ≤ max 2kyp0 k, kyk + and
xp = yp.
Up-Down theorem is very useful and its proof could be found in the book of Pedersen [15]
Theorem 1.34. Let A ⊂ B(H) be a C*-algebra acting nondegenerately on a separable
Hilbert space H. For a self-adjoint element x ∈ A00 , there exists sequences of self
adjoint elements xn ∈ A00 , n ∈ N and xkn ∈ A, k ∈ N such that
(1) The sequence xn is decreasing, i.e xn+1 ≤ xn and xn → x in SOT.
(2) Each sequence {xkn }k∈N for each n ∈ N is increasing, i.e xk+1
≥ xkn and moren
over, xkn → xn for each n as k → ∞ in SOT.
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1.11. Topologies on B(H) and U(H). On a von Neumann algebra, we have six
important locally convex topologies, i.e SOT, WOT, *-SOT, ultrastrong topology,
*-ultrastrong topology and ultraweak topology. In general, these are different unless
the Hilbert space is of finite dimensional (note that the unit ball of B(H) is WOTcompact). An easy fact shows that the ajoint operation is continuous in the WOT
or ultraweak topologies, but not in SOT or ultrastrong topology. This is why we
need to have the SOT-* and ultrastrong-* topologies if we need continuity of the
adjoint operation. I will try to compare these notions with other notions in general
Banach space theory. Recall that, on X ∗ where X is a Banach space, the weak-*
topology is the weaker than the weak topology on X ∗ and the latter is weaker than
the norm topology. The ultraweak topology on a von Neumann algebra M is exactly
the weak-* topology coming from its predual M∗ . Although the WOT topology is
not quite as the analog of the weak*-topology or weak-topology on a Banach space,
the unit ball of M is always WOT-compact (as well as ultraweak compact) while it
is not SOT-compact. This can be proved directly by applying the Banach-Alaoglu
theorem and the fact that restrictions of ultraweak and WOT topologies are the same
on normed-bounded subsets of B(H). Moreover, WOT is weaker than the ultraweak
topology and in fact, it is the weakest topology of common topologies on B(H). The
ultraweak topology can be defined from WOT on B(H) as the following way: We
take any separable infinite dimensional Hilbert space such as `2 . Take the injective
homomorphism φ such that for any T ∈ B(H), φ(T ) is the tensor product operator
T ⊗ id ∈ B(H ⊗ `2 ). The ultraweak topology is the restriction of the WOT on
B(H ⊗ `2 ) onto B(H) via φ. This way also provides us to define ultrastrong and
ultrastrong* topologies coming from the SOT, *-SOT respectively. Also, a useful
and simple criterion between SOT and WOT is that a net Tα converges to T in SOT
if and only if the net (Tα − T )∗ (Tα − T ) converges to 0 in WOT (use polarization).
The SOT and SOT-* coincide on the subset of normal operators ([8]), but otherwise
they are different if we take the net of {S n } where S is a unilateral shift, then (S ∗ )n
converges in SOT to 0 while S n is isometric. Note that the strong-* topologies is used
rather frequently in the theory of Hilbert C*-modules, i.e the strict topology on L(E),
E is a Hilbert C*-module over a C*-algebra. In particular, this strict topology plays
an important role in analysis related to the multiplier of a non-unital C*-algebra.
Next, we summary easy but fundamental facts about behaviors of these topologies
under some certain operations. First, the left and right multiplication with a fixed
bounded linear operator are continuous under these six topologies. However, jointly
continuity of multiplication is more complicated. This operation is continuous in
any of these six topologies. Indeed, even when we restricting on norm-bounded
subsets of B(H), this operation is not WOT-continuous. Consider the unilateral
shift S, then both the sequences S n and (S ∗ )n converge to 0 in WOT but their
product (S ∗ )n S n converges to 1 in WOT. Another reason is that the WOT-limit of a
monotone sequence of projections is not necessarily a projection, but of course this is
true in SOT. Fortunately, is is continuous in SOT, SOT-*, ultrastrong, ultrastrong-*
topologies when we restrict on norm-bounded subsets by triangle inequalities. It is
also useful to note that by the Uniform Boundedness Principle, given any sequence
{Tn } in B(H) such that Tn converges to T in WOT (or any other topologies) then this
10
MINH KHA
sequence is normed-bounded, i.e supn {kTn k} < ∞. Now back to SOT, suppose the
sequence Tn converges in SOT to an operator T then given any holomorphic function
f defined on a neighborhood of the closed unit disk {z ∈ C : |z| ≤ supn kTn k},
we have f (Tn ) → f (T ) in SOT, i.e f is strongly continuous (By induction, we can
easily prove for the case f is a polynomial because of the continuity of multiplication
when restricting to normed-bounded subset with respect to SOT, and then apply the
holomorphic functional calculus). This is not true for WOT by the previous lines
(i.e f (z) = z 2 ). If we extract from the proof of Kaplansky density theorem, any
function f : R → R is strongly continuous on B(H)sa if lim sup|t|→∞ |f (t)/t| < ∞.
An interesting question is whether the WOT convergence implies SOT convergence?
An immediate case is when we have a monotone increasing (or decreasing) net of
positive operators Ti such that Ti converges to T weakly then Ti → T in SOT by
Vigier’s theorem. A less obvious case is the positive cone of the closed unit ball of
B(H). Indeed, let Si be a bounded net of positive operators converging weakly to
1/2
0 then Si → 0 strongly. In fact, let Ti = Si then Ti∗ Ti → 0 in WOT and hence
Ti → 0 in SOT, thus Ti∗ Ti → 0 in SOT since kTi∗ Ti ξk ≤ supi kTi∗ k.kTi ξk. We end this
paragraph by a simple application of the Borel functional calculus and the ultrastrong
topology:
Proposition 1.35. Let x ∈ B(H) be a normal operator. Let {fn } be a bounded
sequence in the space of bounded Borel functions defined on σ(x) such that fn → f
pointwise. Then fn (x) → f (x) in the ultrastrong topology. Moreover, if the sequence
fn ≥ 0 is monotone increasing then fn (x) % f (x) ultrastrongly.
In a nutshell, the restrictions of SOT, *-SOT, WOT coincide with ultrastrong,
ultrastrong-*, ultraweak topologies respectively on normed boudned subsets of M .
Furthermore, an analog of the Banach-Mazur theorem in Banach spaces, which state
that the normed closure and the weak closure of a convex subset are the same, is that
the closures in SOT, WOT, SOT-* topologies are the same for convex subsets of M
while the closures of the ultrastrong, ultraweak, ultrastrong-* topologies are the same
for convex ones too. In particular, when we restrict on norm-bounded, self-adjoint and
convex subsets, the closures in these topologies coincide. This applies in particular
to any self-adjoint subalgebras of B(H), and hence a von Neumann algebra. The fact
on convex subsets could be proved easily if we apply the Hahn-Banach theorem (see
the below proposition) and the following theorem
Theorem 1.36. Let M be a von Neumann algebra on a Hilbert space H. Then a
linear functional f on MPis ultraweak continuous if and
P∞only if there exists {ξn }, {ηn }
∞
in H such that f (x) = n=1 hxξn , ηn i, ∀x ∈ M and n=1 kξn kkηn k < ∞. Moreover,
a linear functional f on M is WOT-continuous if and only if the same happens but
with a restriction that only finite number of ξn , ηn are nonzero.
Proposition 1.37. Suppose that (X, τi ), i = 1, 2 are two locally convex topological
vector spaces with the same algebraic vector space X such that if T is a linear mapping
on X then T is a continuous map in X, τ1 if and only if T is continuous in (X, tau2 ).
Then for any convex subset C ∈ X, the closures of C in τ1 and τ2 coincide.
PERSONAL NOTES IN OPERATOR ALGEBRAS
11
The above theorem is proved in [16] or [9]. A quick corollary is that for a linear
functional on M , the SOT, *-SOT, WOT continuity are the same and the ultrastrong,
ultrastrong-*, ultraweak continuity are the same too. For a normed space (or locally
convex spaces) X, any linear functional T on X ∗ is weak-* continuous if and only
if there is an element x ∈ X such that T (x∗ ) = xast (x), ∀x∗ ∈ X ∗ . Using this fact,
we can prove the first part of the above theorem in the special case M = B(H)
where H is separable: There is an element T ∈ L1 (H) such that f (x) = T r(T x) and
since T is of trace class, T r(|T |) < ∞. Choosing any orthormal basis {ei } on H, we
1/2
1/2
set
P∞ξn = U |T | en , ηn = |T | en , where T = U |T | in the polar decomposition then
n=1 kξn kkηn k ≤ T r(|T |).
We can rephrase the above theorem in this setting
Theorem 1.38. Let M be a von Neumann algebras and φ : M → C be a bounded
linear functional. The following are equivalent:
P
(1) There exist {xn }, {ηn } ∈ H ⊗ `2 such that φ(T ) = n hT ξn , ηn i.
(2) φ is ultrastrong-continuous
(3) φ is ultraweak-continuous
(4) φ is normal (monotone convergence), i.e for any bounded increasing net xi , we
have φ(limn xi ) = limi φ(xi ). Of course, we can replace the limit by supi xi in both
sides.
(5) φ is σ-additive:
For any family of pairwise orthogonal projections (pi ) we have
P
P
φ( i pi ) = i φ(pi ).
Definition 1.39. A bounded linear mapping between two von Neumann algebras
M and N is normal if it is ultraweak to ultraweak continuous or satisfies the fourth
condition (monotone convergence).
We have encountered the point-weak, point-norm, point-ultraweak topologies in
previous subsections on nuclearity, exactness etc. Here is a useful lemma which appeared in proving semidiscreteness of the double dual of a C*-algebra yields nuclearity
of that C*-algebra:
Lemma 1.40. Let A be a Banach space and B(A) be the space of all bounded linear maps from A to A and let C ⊂ B(A) be any convex subset. Then the closures
of C in the point-norm and point-weak topologies are the same. Thus, in order to
prove nuclearity, we can prove idA belongs to the point-weak closure of the convex set
consisting of factorable maps in B(A) as defined in [3].
By the same proof, if M is a von Neumann algebra, then the closures of a convex
subset C ⊂ B(M ) in the point-ultraweak topology and the point-ultrastrong (or pointultrastrong-*) topologies coincide and also, the closures of C in the point-SOT, pointSOT-*, point-WOT topologies are the same. To end this part in this subsection, we
conclude that the main key idea in proving semidiscreteness of A∗∗ implies nuclearity
of A is that roughly speaking, the restrictions on A of all the maps in the closure
in the point-ultraweak topology (or point-weak* topology if you prefer) of the set of
factorable maps of A∗∗ could all be approximated in the point-weak topology from
the set of factorable maps of A. I guess this idea coming from the basic fact in Banach
12
MINH KHA
space theory: the restriction of the weak-* topology on the double dual X ∗∗ when
restricting on X is the same as the weak-topology on X.
We will state a few basic properties of these topologies. These topologies are
metrizable if and only if the Hilbert space H is of finite dimensional. But one can look
at normed-bounded subsets and ask which one of these topologies can be metrizable
in this case? We know that if a normed space X is separable then the unit ball of its
dual X ∗ is weak-* metrizable. The answer to our question is yes due to this analog.
If H is separable, then the unit ball of B(H) is metrizable in these six topologies.
This fact is true if we replace B(H) by any von Nemann algebra whose predual
is separable in its norm topology (We will come back to separable predual of von
Neumann algebras in the next subsection). Another important fact is that the closed
unit ball of B(H) is complete in all these topologies. B(H) is not weakly or strongly
complete, however:
Theorem 1.41. B(H) is complete in the ultrastrong and ultrastrong-* topologies:
Proof. Let (xi ) be an ultrastrong-Cauchy net in B(H). For each compact operator
y ∈ K(H), we have xi y is a norm Cauchy net. So we define L(y) to be the limit. Hence
L(xy) = L(x)y, ∀x, y ∈ K(H), i.e L is a left centralizer for K(H). By Pedersen’s
theorem (II.7.3.6 in [2]), there exists uniquely an element x ∈ B(H) such that L(y) =
xy, ∀y ∈ K(H). This implies xi y → xy, ∀y ∈ K(H). In other words, xi → x
ultrastrongly. If in addition, xi is an ultrastrong*- Cauchy net then apply the above
proof for x∗i , we get the conclusion for ultrastrong-*.
Also, given a von Neumann algebra M , then the closed unit ball M1 is complete
in the SOT and other topologies by the fact we know on B(H). The reverse is also
true, i.e if the closed unit ball of a C*-algebra is complete in SOT then by Kaplansky
density theorem, it is a von Neumann algebra. When dealing with finite von Neumann
algebras, it is more useful to use the norm induced by its faithful normal tracial state.
On bounded sets, the SOT and this norm topology coincide. So we can rephrase the
above statement as the following theorem, which is essential in Haagerup’s proof of
the tracial ultraproduct of von Neumann algebra is still a von Neumann algebra (We
will prove this later):
Theorem 1.42. Let A ⊂ B(H) be a unital C*-algebra equipped with a faithful normal
tracial state τ . Then A is a von Neumann algebra if and only if the closed unit
ball of A is complete in the norm k.kτ which is defined via the formula kxkτ =
τ (x∗ x)1/2 , ∀x ∈ A.
Next, we will study these topologies on the unitary group of B(H), i.e U(H).
Proposition 1.43. On the unitary group U(H), all six topologies are the same and
make U(H) into a topological group.
Proof. First, for SOT and SOT-*, these coincide immediately on any subsets of
normal operators and hence on U(H). Since U(H) is bounded, the ultrastrong,
ultrastrong-* coincide with SOT and the ultraweak coincides WOT too. Hence, it
suffices to prove SOT and WOT coincide on U(H). If (Uλ ) is a net in U(H) such that
(Uλ ) converges in WOT to an element U in U(H) then k(Uλ − U )∗ (Uλ − U )k → 0
PERSONAL NOTES IN OPERATOR ALGEBRAS
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in WOT if and only if 2 − U ∗ Uλ − Uλ∗ U → 0 in WOT. Since adjoint and separate
multiplication are WOT-continuous, we have U ∗ Uλ → 1, Uλ∗ U → 1 in WOT. This
proves (Uλ − U ) → 0 in SOT.
A warning here is that the closure of U(H) in B(H) is not the same in the strong,
weak and strong-* topologies. The strong closure of U(H) consists of isometries,
while the strong-* closure of U(H) is itself because jointly multiplication is SOTcontinuous on bounded subsets. The weak operator closure of U(H) is the whole
unit ball, and thus it contains all coisometries. From here, we can see the unilateral
shift S belongs to the strong closure, but not the strong-* closure while the adjoint
S ∗ belongs to the weak closure, but not the strong closure. An immediate corollary
of 1.43 is:
Corollary 1.44. All six topologies coincide on the set of projections on H.
Proof. There is a one-to-one correspondence between the set of projections and the
set of symmetries (i.e self-adjoint unitaries): P ↔ U = 2P − I.
Note that the set of projections on H is strongly closed as we discussed in paragraph
2 while the weak closure is the positive cone of the closed unit ball.
We end our discussion about U(H) by a beautiful theorem of Dixmier-Douady in
1963:
Theorem 1.45. Let H be an infinite dimensional Hilbert space (maybe nonseparable)
then U(H) is contractible in weak/strong/strong-* topologies while the closed unit ball
of the Hilbert space H is contractible in the norm topology.
Moreover, U(H) is also contractible in the norm topology due to Kuiper (1965) and
more general, it is true for the unitary group of any properly infinite von Neumann
algebra (1976) and of the multiplier of the stable σ-unital C*-algebra M (H ⊗ A)
by Connes-Higson (1987). Note that for finite dimensional case, it fails since the
algebraic topology of U(n, C) is not simple.
1.12. Traces and Conditional expectations in finite von Neumann algebra.
A von Neumann algebra is finite if every projection is finite in the sense of Murrayvon Neumann equivalence. The existence of a faithful normal centre valued trace
implies finiteness of the von Neumann algebra and vice versa. Given such a finite
von Neumann algebra, there exists a faithful family of (scalar) traces by composing
the centre-valued trace with positive elements in the predual of the centre of the von
Neumann algebra. In the case that the finite von Neumann algebra acting on a separable Hilbert space (or the separability in norm
P −iof the predual), there exists a faithful
normal trace since we simply take the sum i 2 h.ei , ei i where {ei } is an orthonormal
basis of the Hilbert space and compose with our centre-valued trace. So under separable predual and finiteness condition, it is safe to assume there is a faithful normal
trace. When the von Neumann algebra is a finite factor, we get the same conclusion
without any separability assumption. Moreover, without separability, there could be
no single faithful normal trace. A simple example is M = `∞ (S), H = `2 (S), where
S is any uncountable set since then this trace is in the predual `1 (S) and hence, it
cannot be faithful since any element of this predual contains uncountable zero-entries.
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MINH KHA
Not so surprisingly, we could expect that the family of II1 factors is large enough
to contain all finite von Neumann algebra. However, its proof may not be so trivial
as it uses the free product construction. For generality, we first state the following
proposition which is due to Ken Dykema in [7]. We will come back to the topic
of simplicity and stable rank of reduced free product of unital C*-algebras later in
optional topics in C*-algebras.
Proposition 1.46. Let Ai be two unital C*-algebras with states φi whose GNS representations are faithful. Let (U , φ) = (A1 , φ1 ) ∗ (A2 , φ2 ). Suppose that the centralizer
of φ1 , which is the set {a ∈ A1 : φ1 (ax) = φ1 (xa), ∀x ∈ A1 }, contains a unital diffuse
abelian subalgebra and also A2 is not isomorphic to C. Then
(1) For any x ∈ U ; ∀ > 0 there exists unitaries z1 , . . . , zn ∈ U for some n ∈ N
such that
n
1X
kφ(x)1 −
zi xzi∗ k < (2)
n i=1
(2) U is simple.
(3) If φi are traces then φ is the unique tracial state on U . If one of φi is not a
trace then U has no tracial state.
If a von Neumann algebra M admits a faithful normal tracial state τ and assume
that this is the unique normal trace on M , then it is a finite factor. Indeed, let P
be a non-zero central projection in M then let π be the normal *-homomorphism
obtained by cut down P on M . Consider the normal tracial state α(τ ◦ π) where
α = (τ (p))−1 > 0 then by uniqueness, this coincides with τ on M . Therefore,
τ (1 − p) = ατ ◦ π(1 − p) = 0. Faithfulness of τ implies p = 1, which shows that M
is a factor.
The following corollary is taken from Proposition 8.1 of [1]:
Corollary 1.47. Let (Ai , τi ) be two finite von Neumann algebras with faithful normal
tracial states τi , i = 1, 2. Let (A, τ ) = (A1 , τ1 )∗(A2 , τ2 ) be the free product of von Neumann algebras. If A2 is diffuse and A1 is not C then A is a II1 -factor. Consequently,
any finite von Neumann algebra with a faithful normal trace can be trace-preserving
embedded inside some II1 -factor.
Proof. Let B be the C*-algebra generated by Ai in A. Then it is clear that B is the
reduced free product of two C*-algebras Ai . Since the centralizer of φ2 is equal to
A2 , it must contain a diffuse masa. By the above proposition, B has a unique tracial
state. Since B is SOT-dense in A, A has a unique normal tracial state. Since A1
is not isomorphic to C, A is infinite dimensional. Hence, A is a II1 factor by the
above observation. Actually, we can deduce factoriality of A from its simplicity as a
C*-algebra.
Next, we consider a rather general and useful situation on the conditional expection. Let M be a finite von Neumann algebra with a faithful normal trace τ . For
any von Neumann subalgebra N ⊂ M , there exists a unique trace preserving normal
conditional expectation EN from M onto N such that
τ (EN (x)y) = τ (xy), ∀x ∈ M, y ∈ N
(3)
PERSONAL NOTES IN OPERATOR ALGEBRAS
15
. EN is a N − N bimodule map. To prove the uniqueness, we do not need to assume
it is normal and hence there exists a unique trace-preserving conditional expectation
such that (3) satisfies.
If N is an injective von Neumann subalgebra of M , there is some connection
between the conditional expectation from B(L2 (M )) onto N 0 and EN 0 ∩M . In fact,
there is a conditional expectation ΨN from B(L2 (M )) onto N 0 which is proper in the
sense that
∀x ∈ B(L2 (M )),
ΨN (x) ∈ conv w {uxu∗ : u ∈ U(N )} ∩ N 0
(4)
It follows that ΨN (x) = x, ∀x ∈ M 0 . Moreover, take any x ∈ M, ΨN (x) ∈ N 0 ∩ M
since U(N ) ⊂ M . It is clear that τ (ΨN (x)y) = τ (xy), ∀x ∈ M, y ∈ N 0 ∩ M since
∀u ∈ U(N ), τ (uxu∗ y) = τ (uxyu∗ ) = τ (xy) and τ is ultraweakly continuous on M .In
other words, ΨN is a trace-preserving conditional expectation from M onto N 0 ∩ M
and hence by the above paragraph, it is EN 0 ∩M on M . Thus
ΨN (xby) = xΨN (b)y,
and in particular,
∀x, y ∈ M 0 ; b ∈ B(L2 (M ))
ΨN (ax) = EN 0 ∩M (a)x, ∀a ∈ M, x ∈ M 0
(5)
(6)
1.13. Some elementary properties of ultrafilter, ultraproduct. First, we recall the correspondence between the space of ultrafilters on N and the Stone-Cech
compactification βN.
Definition 1.48. Denote βN to be the space of all ultrafilters on N. We embed N
into βN via the following map n ∈ N → (n) = {A ⊂ N : n ∈ A}, i.e the principal
ultrafilter containing nN.
Our goal in this part is to equip βN a suitable topology so that βN is exactly the
Stone-Cech compactification of N. For every A ⊂ N, let à = {U ∈ βN : A ∈ U}.
Then it is clear that the collection {Ã : A ⊂ N} forms a base for a topology on βN
˜ B = Ã ∩ B̃) and also, the subspace topology on N is discrete since any
(Ñ = βN, A ∩
ultrafilter contains n must coincide with (n). For two different ultrafilters U 6= V,
there must be a subset A ⊂ N such that A ∈ U, Ac ∈ V and à ∩ Ãc = ∅ since there is
no ultrafilter containing ∅. This implies this topology on βN is Hausdorff. Moreover,
it is also compact. In fact, for any open cover γ of βN, β = {Ã : ∃V ∈ γ : Ã ⊂ V } is
also an open cover of βN by definition of base in topology and so it suffices to a find
a finite subcover β1 of β. Now consider δ = {A ⊂ N : Ã ∈ β}. So ∪A∈δ A = N. If
we assume δ has no finite subcover, then it means G = {N \ A : A ∈ δ} is a centred
system, i.e any finite intersection of members in G is non-empty. So we can find an
ultrafilter U such that G ⊂ U. This implies N \ A ∈ U or δ ∩ U = ∅ or U ∈
/ Ã, ∀A ∈ δ.
Equivalently, U ∈
/ βN (contradiction!). This shows that δ has a finite subcover says
Ui , i = 1, . . . , n such that ∪ni=1 Ui = N. Then {Ũi }i=1,...,n is a finite subcover of β since
any ultrafilter contains one of the sets Ui . Moreover, N is dense in βN. For this, it
suffices to check with non-empty open subsets of the form à where A ⊂ N and in
this case, we always get à ∩ N 6= ∅ since if A 6= ∅, (n) ∈ à when n ∈ A. Another
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MINH KHA
important characterization of Stone-Cech compactification we need to check is that
given any mapping f : N → X, where X is a compact Hausdorff space. We want to
show that f can be extended uniquely to a continuous function f˜ : βN → X. For the
existence of this map, let U ∈ βN be given. Let f∗ (U) be the ultrafilter on X defined
by the relation A ∈ f∗ U ⇔ f −1 (A) ∈ U. Since X is compact, there exists a limit
lim f∗ U := limn→U f (n) = y in X. Set f˜(U) = lim f∗ U. It remains to prove continuity
of f˜. Fix an open subset U ⊂ X, consider any element U such that f˜(U) ∈ U we
want to find a neighborhood V of U such that f˜(V ) ⊂ U . Using regularity of X,
we find a neighborhood W of f˜(U) such that W ⊂ W̄ ⊂ U . Let A = f −1 (W ). The
condition f˜(U) ∈ W means {n ∈ N : f (n) ∈ W } ∈ U or A ∈ U and in particular, A
is nonempty. Our claim is that f˜(Ã) ⊂ U . Assume not, there exists one ultrafilter
V containing A such that f˜(V) ∈
/ W̄ . The latter condition means f −1 (W̄ c ) ∈ V
and so f −1 (W c ) ∈ W which is a contradiction since A ∈ V ! So f˜ is continuous.
The uniqueness of f˜ is deduced from a general statement: If f, g are two continuous
functions from a topological space A to a Hausdorff space B such that f = g on a
dense subset Z of A then f = g. In our setting, B is the compact Hausdorff space
X.
Let U be a non principal ultrafilter on N. Given a sequence of Banach spaces {X k },
we can form the ultraproduct of these Banach spaces, i.e X U by taking the quotient
`∞ ({X k })/NU where NU is the closed normed subspace consisting of sequences {xk }
where xk ∈ X k such that limn→U kxk k = 0. The norm on X U is defined by the
formula kxkU = limn→U kxk kX k . Then (X U , k.kU ) is a Banach space. If there exists
some N ∈ N such that {k ∈ N : dim(X k ) ≤ N } ∈ U then dim(X U ) ≤ N .
Given a sequence of bounded linear operators T k : X k → Y k , where X k , Y k
are Banach spaces, such that supk kT k k < ∞. We can define T U : X U → Y U
such that T U (xk ) = (T k (xk )), ∀(xk ) ∈ X U . Moreover, kT U k = limk→U kT k k since
limn→U (xn yn ) = limn→U xn limn→U yn .
A specific case of the above construction is the ultraproduct of a sequence of
Hilbert spaces {Hn }. The inner product on the ultraproduct H U is just the natural
extension by taking the ultralimit process. Given another sequence of Hilbert spaces
Kn , we can form the 2-tensor product (or Hilbert Schmidt norm if you prefer) of
the two ultraproducts H U ⊗2 K U and this Hilbert space is canonically
isometric
Q
n
n
U
U
n
embedded into the ultraproduct of H ⊗2 K , i.e H ⊗2 K ,→ n→U H ⊗2 K n . In
particular, if supk dim(Hk ) < ∞, this turns out to be an isomorphism. Indeed, we
P
n)
just
the surjectivity for elements of the form η = ( dim(H
eni ⊗ yin )n ∈
i=1
Q neednto check
n
n
n
n→U H ⊗2 K where {ei }i=1,...,dim(H n ) is any othornormal basis for H for each n
P
n)
dim(H
and yin ∈ Y n . So supn i=1
kyik k2 < ∞. Put eni = 0, yin if i > dim(H n ) and
P
n
n
U
U
let N = supn dim(H n ). The element N
maps to our
i=1 (ei )n ⊗ (yi )n ∈ H ⊗2 K
element η.
One final remark is that if the metric ultraproduct of separable metric spaces is
separable then it is compact and for any > 0, the sequence of covering numbers
N (Xn , ) is essential bounded, i.e the subset of n such that the sequence is above
bounded is in the ultrafilter U.
Reference: Appendix by Pestov in [4].
PERSONAL NOTES IN OPERATOR ALGEBRAS
17
1.14. The functoriality of reduced group C* algebra and group von Neumann algebra. A natural question arises when we want to see the connection between morphisms of groups and *-homomorphism on the level of reduced group C*
algebra or group von Neumann algebra:
Question: Given a group homomorphism f : G → H, is there any corresponding
∗
∗
*-homomorphism f : Cred
(G) → Cred
(H) such that it extends the original morphism
on groups f ?
In general, we could not always expect any good functoriality as in our question.
In fact, a discrete group G is amenable if and only if the trivial representation 1G
is weakly contained in the left regular representation λG . Hence, for any f ∈ `1 (G),
k1G (f )k ≤ kλG (f )k. So for a non-amenable group G, there is some f ∈ C[G] such that
∗
(G) onto
k1G (f )k > kλG (f )k and thus, we cannot find a homomorphism from Cred
C such that it extends the trivial representation 1G on G. In short, G is amenable
∗
iff Cred
(G) has a character or one-dimensional representation. It turns out that
amenability is exactly the key as in the following theorem
Theorem 1.49. Let f : G → H be a homomorphism between discrete groups. Then
the homomorphism f extends to a homomorphism of reduced group C ∗ -algebras if and
only of ker(f ) is amenable, and extends to a homomorphism of group von Neumann
algebras if and only if ker(f ) is finite.
Proof. We consider the C ∗ -group algebra first. Indeed, given such a map f whose
kernel N is an amenable group, then the trivial representation is weakly contained
in the left regular representation on N and hence by the continuity of induction
operations, we must have the left regular representation on G/N (which is the induced
representation of 1N on G) is weakly contained in the left regular representation on G.
This is equivalent to see that kλG/N (f )k ≤ kλG (f )k, ∀f ∈ `1 (G) and hence the map
∗
∗
∗
∗
from Cred
(G) to Cred
(G/N ) is surjective, and thus a map from Cred
(G) to Cred
(H).
For the converse, note that if there exists such a *-homomorphism of reduced group
∗
C ∗ -algebras, then the restriction of this map onto Cred
(Kerf ) has one dimensional
∗
∗
image C because Cred (Kerf ) ⊂ Cred (G). Therefore, the trivial representation of
Ker(f ) is weakly contained in the left regular representation on Ker(f ) and hence
Ker(f ) is amenable.
An alternative explaination for the case when Ker(f ) is amenable,
a
P we can take
−1/2
2
Folner sequence {Fn } of Ker(f ). We define a vector ξn := |Fn |
g∈Fn δg in ` (G).
∗
The corresponding vector state φn = h.ξn , ξn i on Cred (G) will have a weak limit φ by
Banach-Alaoglu theorem. Since {Fn } is a Folner sequence, an easy computation gives
∗
us that φn (a) → 1Ker(f ) (a), ∀a ∈ C ∗ (Ker(f )) and thus φ = 1Ker(f ) on Cred
(Ker(f )).
Furthermore, the left regular representation on G of an element g ∈ G\Ker(f ) would
move the support of each vector ξ ∈ `2 (Ker(f )) ⊂ `2 (G) away from Ker(f ) and thus
φ = 0 on the closure of the linear span of {λg : g ∈
/ Ker(f )}. In other words, our
state φ is exactly the extension of the characteristic function χKer(f ) on G. By the
uniqueness of GNS construction, the GNS-representation πφ wrt. φ is the extension
∗
on Cred
(G) of the quasi-regular representation: λKer(f ) : G → B(`2 (G/Ker(f )))
∗
which maps each g into λgH . Thus πφ is a ∗-homomorphism from Cred
(G) onto
∗
Cred (G/Ker(f )).
18
MINH KHA
A third proof using the Reiter’s condition of amenability is given in the proof
of Proposition 3.3 in [?JS]. It also boils down to prove the characteristic function
χKer(f ) on G extends to a state φ on the reduced C ∗ -group algebra. If we let µ be
∗
(G), then the composition φ ◦ µ (which
the canonical surjection from C ∗ (G) → Cred
is ωχKer(f ) ) is a state on the full group C ∗ -group algebra of G. Hence, the GNS
representation πφ◦µ is unitarily equivalent to the composition πφ ◦ µ. This means the
∗
range of πφ is *-isomorphic to the range of πχKer(f ) = Cred
(G/ker(f )).
For the von P
Neumann algebra case, assume first H = Ker(f ) is a finite subgroup,
2
−1
let P = |H|
g∈H λg ∈ B(` (G)) then P is a central projection in L(G). Indeed,
it is clear that P is a projection since H is a finite subgroup. Since H is also normal
in G, P is central in L(G). Our claim is that the corner P L(G)P is L(G/H) since
then the mapping we need is just the restriction of the cut-down map by the central
projection P .
For the reverse, it is easy to see that by the previous section that there is a rank
one projection P ∈ L(H). If H is an infinite discrete group, L(H) ∩ K(`2 (H)) = {0}.
This impliesP
P ∈
/ L(H). Hence, H is finite. In this case, we can easily prove that
−1
P = |H|
g∈H λg by using the conditions that P = θx,x for some unit vector
P
x = h∈H ah δh and [P, λh ] = 0, ∀h ∈ H.
Remark 1.50. Note that given a positive type function φ on a discrete group G, we
can associate a corresponding state ωφ on C ∗ (G), but not always this construction
∗
yields a state on Cred
(G). However, we can always associate to a u.c.p multiplier mφ
∗
on both L(G), Cred (G), C ∗ (G) by Fell-absorption principle.
Corollary 1.51. Let H be a normal subgroup of a discrete group G. Then there is
∗
∗
a *-homomorphism ρ : Cred
(G) → Cred
(G/H) that extends the canonical surjection
ρ : G → G/H iff H is amenable.
In general, we could expect the multiplier algebra of the reduced group C ∗ -algebra
is large enough to get such a functoriality.
Theorem 1.52.
All resources here I took from answers of Matthew Daws, Andrea Thoms and Alain
Valette on Mathoverflow.
2. Some Topics in Operator Theory
2.1. On the noncommutative Weyl-von Neumann theorems by Voiculescu.
This subsection I follow from the presentation of Davidson in his standard book [6].
First, we recall the Weyl-von Neumann-Berg theorem , which is important itself.
There are some weaker notions of equivalence that are important. We say two
operators A and B are approximately unitarily equivalent (write A ∼a B) if there
is a sequence of unitary operators Un such that B = limn→∞ Un AUn∗ . This notion
is equivalent to say that A and B have the same norm-closed unitary orbit Ū(A) =
cl{U AU ∗ , U ∈ U(H)}.
Two operators A and B are approximately unitarily equivalent relative to K (write
A ∼K B) if in addition to having B = limn→∞ Un AUn∗ , one also has that B −Un AUn∗ ∈
K.
PERSONAL NOTES IN OPERATOR ALGEBRAS
19
Two representations σ and ρ of a separable C*-algebra A are approximately unitarily equivalent (relative to K) if there is a sequence of unitary operators such that
ρ(A) = limn→∞ Un σ(A)Un∗ for all A ∈ A (and in addition the range of ρ − AdUn σ ∈ K
for every n ≥ 1). Again we write σ ∼a ρ and σ ∼K ρ respectively.
Two representations σ and ρ of a separable C*-algebra A are weak approximately unitarily equivalent (write σ ∼wa ρ) if there are two sequences Un , Vn of
unitary operators such that σ(A) = W OT − limn→∞ Un ρ(A)Un∗ and ρ(A) = W OT −
limn→∞ Vn σ(A)Vn∗ . Note that in general, Un 6= Vn . Moreover, this notion is equivalent to another one which looks at first stronger, i.e weak approximately unitary
equivalence between two representations of a C*-algebra yields strong*-approximately
unitary equivalence also.
Two operators A and B are weak approximately unitarily equivalent (write A ∼wa
B) if there are two sequences Un , Vn of unitary operators such that A = SOT ∗ −
limn→∞ Un BUn∗ and B = SOT ∗ − limn→∞ Vn AVn∗ . This implies the natural ∗isomorphism ρ : C ∗ (A) → C ∗ (B) is weak approximately unitarily equivalent to the
identity representation idC ∗ (A) .
Here is out starting point
Theorem 2.1. Let A be a separable abelian C*-algebra. Given any finite collection
of projections X in A”, then there exists an orthonormal basis {en } in H such that
D + K contains A, where D is the algebra of all diagonal operators relative to the
basis {en } such that X ⊂ D.
Proof. First, the abelian C*-algebra A is contained in a singly generated abelian C*algebra C ∗ (A) for some self adjoint operator A. Thus, A ⊂ C = cl(span(Θ)), where
Θ = {En } ∪ X . Here En ∈ W ∗ (A) are spectral projections of A and we suppose that
X are the first N elements of Θ. Therefore, Θ is a countable collection of commuting
projections.
The next goal is to construct an approximate unit of projections for K which is
quasicentral for C as follows. Fix an orthonormal basis x1 , x2 , . . . for H. For each
projection Ei , we denote Ei−1 = 1 − Ei and Ei1 = Ei . Then for k ≥ N , we set
Q
Lk = span{ ki=1 Eii xj , 1 ≤ j ≤ k, i ∈ {−1, +1}}. Then Lk is an increasing sequence
of finite dimensional subspaces with dense union since {x1 , . . . , xk } ⊂ Lk . Let Fk be
the orthogonal projection onto Lk . Then Fk converges strongly to I, and hence this
sequence is an approximate unit for K. By commutativity, each Lk is invariant for
En whenever n ≤ k. Therefore, limk→∞ Fk En −En Fk = 0 and this conclusion extends
to the closed linear span, i.e C.
Let Dn = En when n ≤ N and Dn = En (1 − Fn ) when n > N . Let D be the
C*-algebra generated by {Dn }. Then since Dn commutes with other Dm ’s, D is an
abelian C*-algebra such that A ⊂ C ⊂ D + K since En = Dn + En Fn ∈ D + K for
n > N.
The final goal is to show that D is diagonalizable. Since Dn commutes with every
Fk for k ≥ N , the finite dimensional subspaces LN and Lk Lk−1 for k > N are all
invariant subspaces for D. By finite dimensional spectral theorem, the restriction of
D onto each those invariant subspaces is a commuting family of normal matrices and
hence it is diagonalizable simutaneously. We choose such a basis for each of these
20
MINH KHA
subspaces and since they are orthogonal subspaces of H, we just combine these bases
for each k to get an orthonormal basis {en } so that D is diagonalizable relative to
this basis.
Next we state the Weyl-von Neumann-Berg’s theorem which asserts that every
normal operator is a small compact perturbation of a diagonalizable operator.
Corollary 2.2. Every normal operator N on a separable Hilbert space can be expressed as a sum N = D + K of a diagonal normal operator D and a compact
operator K. Moreover, for any > 0 and any n commuting Hermitian operators
A1 , . . . , An , there are simultaneously diagonal Hermitian operators Di and compact
operators Ki such that Ai = Di + Ki and max1≤i≤n kKi k ≤ .
Proof. First, we can translate each Ai by a multiple of the identity and scale them
so that 0 ≤ Ai ≤ I. Then we need to rewrite each Ai as an approriate linear
sum of its spectralP
measures EAi . The goal is to choose such a nice expression for
−k
Ai such as Ai =
k≥1 2 EAi (Bk ) for some subsets Bk ⊂ [0, 1]. It’s clear that
Pk
Ai = limk→∞ 2j=0−1 2−k EAi ((j2−k , 1])). Since we want the above form, we should
P2k −1 −k
P k+1
2 EAi ((j2−k , 1])) or equivhave 2−k EAi (Bk ) = 2j=0 −1 2−k EAi ((j2−k−1 , 1]))− j=0
k−1
alently, Bk = ∪2j=1 (2−k (2j − 1), 2−k (2j)]). Choose N large enough so that 2−N < .
Then applying the previous proof to the family Θ = {EAi (Bk ) : k ≥ 1, 1 ≤ i ≤ n}
(i)
to obtain a diagonal algebra D which contains Dk = EAi (Bk ) for 1 ≤ k ≤ N and
(i)
1 ≤ i ≤ n and for each other Dk = EAi (Bk ) − Rik where Rik is a finite rank proP
(i)
jection. Then let Bi = k≥1 2−k Dk then each Bi is a positive diagonal contraction
in D and the differences Ki = Ai − Bi are compact operators whose norms are less
or equal than 2−N < as we wished. For general normal operator A, we apply this
result to its real and imaginary parts.
This lemma was proved in Davidson’s book. We skip its proof here.
Lemma 2.3. Suppose that X is a compact metric space. Let {ξk : k ≥ 1} and
{ζk : k ≥ 1} be two countable dense subsets of X such that each isolated point of
X is repeated the same number of times in each sequence. Then given any >
0 there is a permutation π of N so that dist(ξk , ζπ(k) ) < for any k ∈ N and
limk→∞ dist(ξk , ζπ(k) ) = 0.
Theorem 2.4. Suppose that M and N are normal operators on separable Hilbert
spaces then M ∼a N (relative to K) if and only if σe (M ) = σe (N ) and ker(M −λI) =
ker(N − λI) for all λ ∈ C \ σe (M ).
Proof. Suppose that both M, N are approximately unitarily equivalent normal operators. If Un are unitary operators such that N = limn→∞ Un M Un∗ then it is obvious
that X = σe (M ) = σe (N ) and f (N ) = limn→∞ Un f (M )Un∗ for all f ∈ C(X).
The Weyl-von Neumann-Berg Theorem could be viewed under another point view
in terms of representations of C(X). A representation is called diagonal if its range
is diagonalizable.
PERSONAL NOTES IN OPERATOR ALGEBRAS
21
Corollary 2.5. Let X be a compact metric space. Then every representation of
C(X) on a separable Hilbert space is approximately unitarily equivalent to a diagonal
representation relative to K.
Here is the classification theorem on approximate unitary equivalence of representations of abelian C*-algebras C(X) for compact metric spaces X.
Theorem 2.6. Let X be a compact metric space. Suppose that ρ and σ are separable
representations of C(X). Then the following statements are equivalent:
(i) ρ ∼K σ
(ii) ρ ∼a σ
(iii) ρ ∼wa σ
(iv) rank(ρ(f )) = rank(σ(f )) for all f ∈ C(X).
2.2. Quasitriangular Operators. For this subsection, I follow very tightly to the
presentation of Carl Pearcy in his book [14]. We begin with some preliminaries on
weak and strong convergence in B(H).
Proposition 2.7. If {An }, {Bn } are two sequences in B(H) that converge weakly and
strongly respectively to operators A0 , B0 then the sequence {An Bn } converges weakly
to A0 B0 . Furthermore, if {xn } is a sequence of vectors in H such that kxn − x0 k → 0
then {An xn } converges weakly to Ax0 .
Proof. For the first assertion, we note that since the functionals Zn (x) = hAn x, yi converges to Z0 (x) = hA0 x, yi, it follows by Banach-Steinhaus theorem that supn kA∗n (y)k =
supn kZn k < ∞. By triangle inequalities, we easily get hAn Bn x−A0 B0 x, yi → 0. The
second assertion follows immediately from the fact that supn kA∗n y − A∗0 yk < ∞. Here is an elementary proposition about compact operators.
Proposition 2.8. For an operator T ∈ B(H), the following statements are equivalent:
(1) T ∈ K(H)
(2) For every bounded net {xλ } of vectors in H, the net {T xλ } has a strongly
convergent subnet.
(3) If {xλ } is any bounded net such that it converges weakly to a point x0 then the
net {T xλ } converges strongly to T x0 .
(4) If {xn } is any sequence of vectors in H converging weakly to a point x0 then
kT xn − T x0 k → 0.
Proof. We sketch the proof. For (2) implies (3), note that T is weak to weak continuous and the fact that a net in a topological space converges to a point p iff every
subnet of that net has a subsubnet that converges to p. Here p = T x0 . For (4) implies
(1), suppose for contradiction that the image of the close unit ball B of H under T
is not strongly compact then T (B) is not sequentially compact since it is a metric
space. Then we can find a sequence {T xn } where kxn k ≤ 1 that has no strongly
convergent subsequence. Note that B is weakly compact, hence some subsequence
xnk converges weakly to a vector x0 and by (4), it gives a contradiction.
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MINH KHA
Note that the condition in (4) does not need the boundedness assumption since
for sequences, this is automatically satisfied by Banach-Steinhaus theorem. Here
are some useful corollaries related to the effect of left and right multiplications by a
compact operator to the convergence notions (weak to strong for left multiplication
and strong to uniform for right multiplication) in a Hilbert space (we can replace a
sequence by a bounded net in those corollaries)
Corollary 2.9. If K is a compact operator and {An } is a sequence in B(H) that
converges weakly to A then {KAn } converges strongly to KA. In particular, An KAn
converges weakly to AKA.
Corollary 2.10. If K is a compact operator and {An } is a sequence in B(H) that
converges strongly to A then kAn K − AKk → 0.
Proof. Suppose for contradiction, we can find an > 0 and a sequence {xnk } of
unit vectors in H such that k(Ank K − AK)xnk k ≥ . By passing to a subsequence
if necessary, we assume xn converges weakly to x0 , so kK(xnk − x0 )k → 0. From
the inequality k(Ank − A)Kxnk − (Ank − A)Kx0 k ≤ M kKxnk − Kx0 k, we get a
contradiction.
Corollary 2.11. If K ∈ K(H) and {Pn } is a sequence of projections converging
weakly to a projection P then kKPn − KP k → 0. If we drop the condition P is a
projection then we have kPn x − xk → 0 for all those vectors such that x = P x.
Proof. First, we note that since kPn x − P xk2 ≈ hPn x, Pn x − P xi = hPn x, x − P xi, it
follows that Pn converges strongly to P . Now this corollary follows from the preceding
one since we get kK(Pn − P )k = k(Pn − P )K ∗ k.
Corollary 2.12. If {Pn } is a sequence of projections in B(H) converging weakly to
1H and K ∈ K(H) then kPn KPn − Kk → 0.
Proof. It follows directly from this inequality kPn KPn − Kk ≤ kPn kkKPn − Kk +
kPn K − Kk
Now we come to the definition of the quasitriangular operators.
Definition 2.13. An operator T in B(H) is called quasitriangular if there exists
a sequence of projections {Pn } of finite rank in B(H) that converges strongly (or
weakly) to 1H and satisifes kPn T Pn − T k → 0. The class of quasitriangular operators
in B(H) is denoted by QT (H).
We now state some basic structure theorems concerning quasitriangular operators
developed by Halmos and Douglas-Pearcy.
Definition 2.14. An operator T is called triangular if there exists an orthonormal
basis {en } of H such that the matrix (aij ) for T relative to this basis is in upper
triangular form, i.e aij = 0 whenver i > j.
Proposition 2.15. Every triangular operator is quasitriangular.
Proof. We take Pn to be the projection onto the span of {ei }1≤i≤n for each natural
number n. This follows from the fact that the range of each Pn is invariant under
PERSONAL NOTES IN OPERATOR ALGEBRAS
23
T since its matrix relative to the basis ei is of the triangular form and hence it is
equivalent to T Pn = Pn T Pn .
Theorem 2.16. An operator T is quasitriangular if and only if lim inf P ∈P kP T P −
T P k = 0 where P is the directed set consisting of all projections in B(H) of finite
rank.
Theorem 2.17. QT (H) is norm-closed.
This theorem should be expected from the definition of quasitriangularity:
Theorem 2.18. Every normal operator T belongs to QT (H).
Proof. From Berg’s theorem, for each > 0 there exists a diagonal operator D such
that T is approximately unitary equivalent to D modulo a compact operator K such
that kKk ≤ . D is triangular and hence D ∈ QT (H). By definitions and Corollary
(2.12), quasitriangularity is preserved under approximate unitary equivalent relative
to compact operators. Hence T ∈ QT (H).
Proposition 2.19. Every compact operator is quasitriangular. Moreover, if T ∈
QT (H) and K ∈ K(H) and λ ∈ C then T + λ + K ∈ QT (H).
Theorem 2.20. If T is quasitriangular then for every > 0 there exists a triangular
operator T such that T ∼ T (). Hence, the set QT (H) can be characterized as the
set of all sums of the form T0 + K where T0 is triangular and K ∈ K(H).
2.3. On basic properties of index theory in subfactors. We recall an elementary exercise in [9] about the coupling constant of a finite von Neumann algebra M
whose commutant is also finite.
Proposition 2.21. Let M be a finite von Neumann algebra such that M 0 is finite. Let
τ, τ 0 be two unique centre-valued traces on M, M 0 . Then there is a positive real number
c > 0 such that: Given any two projections E, F in M, M 0 respectively then τ (E) =
cτ 0 (F ) if and only if there exists a vector y such that E, F has ranges [M 0 y], [M y]
respectively.
Proof. To prove this, we divide it into three steps:
Step 1:
Assume there is a vector x0 which is generating and separating for M . Then we
find a *-anti isomorphism between M and M 0 . Let’s denote this isomorphism as
φ then φ(A)x0 = Ax0 , ∀A ∈ M . Then it is easy to see that given any projection
E ∈ M , its range is [M 0 Ex0 ] and thus the range of φ(E) = [M Ex0 ]. Note that
τ 0 ◦ φ is also a tracial state on M 0 and so τ (E) = τ 0 (φ(E)). Now assume E, F are
projections in M, M 0 respectively such that τ (E) = τ 0 (F ). Then φ(E) ∼ F . Let
V ∈ M 0 be the partial isometry between φ(E) and F . Let y = V Ex0 . Therefore:
F (H) = V φ(E)H = [V M Ex0 ] = [M y]. Moreover, E = [M 0 Ex0 ] ≥ [M 0 V Ex0 ] =
[M 0 V φ(E)x0 ] ≥ [M 0 V ∗ V φ(E)x0 ] = [M 0 φ(E)x0 ] = [M 0 Ex0 ] = E. This implies E =
[M 0 y] and so we get our desired conclusion. Conversely, if E = [M 0 y], F = [M y] then
[M 0 y] = [M 0 Ex0 ](= E) and so F = [M y] ∼ [M Ex0 ] = φ(E). Thus, τ (E) = τ 0 (F ).
Step 2:
24
MINH KHA
Now assume x0 is a generating vector for a finite factor M . Thus the commutant
M 0 is finite since the projection [M 0 x0 ] in M is finite would imply I = [M x0 ] is finite
in M 0 . Let E = [M 0 x0 ] and c0 = τ (E). We will prove the following statement: Given
two projections F, F 0 in M, M 0 respectively. Then τ (F ) = c0 τ 0 (F 0 ) iff there is a vector
y such that F = [M 0 y], F 0 = [M y].
To prove this, first we notice that if either τ (F ) = c0 τ 0 (F 0 ) or there is a vector y
such that F = [M 0 y], F 0 = [M y] then F . E. For this, first if τ (F ) = c0 τ 0 (F 0 ) then
τ (F ) ≤ c0 = τ (E) and so F . E by properties of the dimension function on M .
Finally, if F = [M 0 y], F 0 = [M y] then F ≤ [M x0 ] = I implies F . [M 0 x0 ] = E.
So we can suppose F . E. Let V be the partial isometry between F and E. The
idea is that we want to work with F0 = V F V ∗ ≤ E instead of F . Thus τ (F0 ) =
τ (F ) = c0 τ 0 (F 0 ). Now the fact that there is a vector y such that F = [M 0 y], F 0 = [M y]
is also equivalent to there is a vector y0 such that F0 = [M 0 y0 ], F 0 = [M y0 ]. We simply
choose y0 = V y. The proof of this is similar to step 1. (Indeed, F 0 = [M y] ≥ [M y0 ]
and [M y] ∼ [M y0 ] since F ∼ F0 ⇒ F 0 = [M y0 ] since M 0 is finite).
Now assume F ≤ E. Let τ1 , τ10 be the tracial states on EM E, M 0 E respectively.
Then τ (EAE) = c0 τ1 (EAE), τ10 (BE) = τ 0 (B), ∀A ∈ M, B ∈ M 0 by the uniqueness
of traces on finite factors. Take A = F, B = F 0 , c0 τ (F ) = τ1 (F ) and τ10 (F 0 E) =
τ 0 (F 0 ) would yield τ1 (F ) = τ10 (F 0 E) is equivalent to τ (F ) = c0 τ 0 (F 0 ). Consider
R = EM E, R0 = M 0 E, x0 is a generating and separating vector for R. Hence, we
can apply Step 1 to F and F 0 E in R and R0 respectively. Then τ1 (F ) = τ10 (F 0 E) iff
there is a vector y such that F = [M 0 Ey], F 0 E = [EM Ey]. Since the central carrier
of E in M is the identity, the map B 0 → B 0 E, B 0 ∈ M 0 is a *-isomorphism between
M 0 and M 0 E. Because E[M Ey] = [EM Ey] = F 0 E, the injectivity of the above map
gives [M Ey] = F 0 , which concludes step 2.
Step 3:
Assume M, M 0 are finite factors. Since a finite von Neumann algebra is countably
decomposable if its center is countably decomposable, factoriality always gives us
countable decomposability of both M and M 0 . In any countably decomposable von
Neumann algebra M , we can find a central cyclic projection P ∈ M such that I − P
is cyclic for M 0 . But factoriality yields P is either 0 or 1. In the former case, M 0
has a generating vector x0 while in the latter case, M has a generating vector x0 .
By combining with step 2, we get our desired conclusion. Note that the coupling
constant c0 = τ 0 ([M x0 ]) in the first case while c0 = τ ([M 0 x0 ]) in the latter case. 2.4. Solidity, Strong Solidity. The remarkable paper of Ozawa [12] is one of the
first fundamental ones to make the concept of solid von Neumann algebras becomes
important in rigidity and deformation theory and other areas. Moreover, this paper
introduces an attractive method of using C*-algebra to study von Neumann algebra.
Liming Ge used entropy techniques in free probability theory to prove that countable
free group factor is prime. The result of Ozawa proves that it is also solid and hence
by one of this theorem, this implies non-property gamma and primeness results.
His starting point in the paper is the theorem of Akemann and Ostrand. The
product map of the left and right regular representations when passing to the Calkin
PERSONAL NOTES IN OPERATOR ALGEBRAS
25
algebra
Cλ∗ (G) ⊗min Cρ∗ (G) → B(`2 (G))/K(`2 (G))
(7)
is continuous if G is the free group F2 . Later, Skandalis extended for all discrete
subgroups of all connected simple Lie groups of rank one and Higson-Guentner proved
for all word hyperbolic groups.
For these groups, the reduced group C*-algebra has property (C) of Archbold and
Batty. By Effros-Haagerup, this implies local reflexivity.
Lemma 2.22. Let G be as above then Cλ∗ (G) is locally reflexive. That means, for any
finite dimensional operator system E ⊂ Cλ∗ (G)∗∗ there exists a net of unital completely
positive maps θi : E → Cλ∗ (G) which converges to identity on E in the point-weak*
topology.
We need the following definitions.
Definition 2.23. A von Neumann algebra M is called solid if for any diffuse sub-von
Neumann algebra A ⊂ M , the relative commutant A0 ∩ M is injective.
Solidity implies finiteness by an easy argument. Indeed, if M is not finite, we can
¯
assume M is properly infinite. Then M is isomorphic to M ⊗B(H)
for a separable
infinite dimensional Hilbert space H. We choose A = C ⊗ L(F2 ) which is diffuse since
¯
A is a II1 factor. The relative commutant of A in M is isomorphic to M ⊗R(F
2 ),
which is noninjective since there is a trace preserving conditional expectation from
¯
M ⊗R(F
2 ) onto R(F2 ) while the latter algebra is noninjective.
Definition 2.24. A von Neumann algebra M satisfies property (AO) if there are
unital ultraweakly dense C*-subalgebras B ⊂ M, C ⊂ M 0 such that B is locally
reflexive and the *-homomorphism
µ : B ⊗ C → B(H)/K(H)
(8)
Pn
which sends i=1 ai ⊗xi to π( i=1 ai xi ), is continuous with respect to the minimal
tensor norm on B ⊗ C.
Pn
Here is the main theorem
Theorem 2.25. If M is finite and does satisfy property (AO) then M is solid.
Proof. Step 1:
Let B ⊂ M ⊂ B(H), C ⊂ M 0 be ultraweakly dense C*-subalgebras such that
B is locally reflexive. Let N ⊂ M be a von Neumann subalgebra with a normal
conditional expectation EN onto N . Suppose that the unital completely positive
map
ΦN :B ⊗ C → B(H)
n
n
X
X
ai ⊗ x i →
EN (ai )xi
i=1
i=1
is continuous with respect to the minimal tensor norm. Then N is injective.
By Arveson extension theorem and the fact that B ⊗min C ⊂ B(H) ⊗min C, we
extend ΦN to a unital completely positive map Ψ : B(H) ⊗min C → B(H). So Ψ
26
MINH KHA
is a C-bimodule map. Let ψ(a) = Ψ(a ⊗ 1), ∀a ∈ B(H). Then ψ(a)x = Ψ(a ⊗
1)Ψ(1 ⊗ x) = Ψ(a ⊗ x) = Ψ(1 ⊗ x)Ψ(a ⊗ 1) = xΨ(a). Hence, Ψ(a) ∈ C 0 = M (since
C 0 = C 000 = M 00 = M ). Let ψ̄ = EN ψ : B(H) → N be a unital completely positive
map such that ψ̄|B = EN |B by definition of ΦN . Now we use local reflexivity of B to
induce conditional expectation from B(H) onto N to deduce N is injective.
Consider the following directed set I = {(E, F, ) : E ⊂ N, F ⊂ N∗ , > 0} whose
the relation ≤ is that (E1 , F1 , 1 ) ≤ (E2 , F2 , 2 ) iff E1 ⊂ E2 , F1 ⊂ F2 and 1 ≥ 2 .
Fix an element i = (E, F, ) ∈ I. Let E be the finite dimensional operator system
∗
generated by the finite subset E. Since M = B 00 = pB ∗∗ , EN
(F) ⊂ M∗ = pB ∗ where p
is the central cover of the representation B ⊂ B(H), there exists a unital completely
∗
∗
positive map θi : E → B such that for a ∈ E, f ∈ F, |EN
(f )(θi (a)) − EN
(f )(a)| < .
By Arveson, we extend θi onto B(H). Let σi = ψ̄θi : B(H) → N . It follows that for
∗
a ∈ E, f ∈ F, |hσi (a), f i − ha, f i| = |hψ̄θi (a), f i − hEN (a), (f )i| = |hθi (a), EN
(f ) −
∗
ha, EN (f )i| < . By Banach-Alaoglu, the net σi of unital completely positive maps
has a cluster point in the point ultraweak topology. Let σ be one of these cluster
points, then hσ(a), f i = ha, f i, ∀a ∈ N, f ∈ N∗ or σ|N = id|N . Of course σ is a
unital completely positive map and thus by Tomiyama theorem, σ is a conditional
expectation as we desire.
Step 2:
Now let A be a diffuse von Neumann subalgebra in M . By passing through a
maximal abelian subalgebra of A, we can suppose that A is abelian, diffuse and it
is enough to prove injectivity of N = A0 ∩ M . By Step 1, it suffices to show the
map ΦN is continuous with respect to the minimal tensor norm on B ⊗ C. We
know that any diffuse abelian von Neumann algebra with separable predual is *isomorphic to L∞ [0, 1]. Hence there exists a unitary u ∈ A such that limk→∞ uk = 0
ultraweakly (the
P Riemann-Lebesgue lemma for the function u(t) = exp(2πit)). Let
Ψn (a) = n−1 nk=1 uk au−k , a ∈ B(H) and choose a cluster point Ψ : B(H) → B(H)
in the point ultraweak topology. Our claim is that Ψ|M = EN by proving it is a
trace-preserving conditional expectation onto N . First, Ψ(a) = a, ∀a ∈ N and it is
also unital completely positive. Also, to prove it maps onto N , it suffices to show
that for every j ∈ N, Ψ(a)uj = uj Ψ(a), ∀a ∈ M since A is generated by u. This
P
P
k
−k
can be seen easily by showing ∀a ∈ B(H), nk=1 uk au−k uj = uj n−j
then
k=1−j u au
P
P
n
n−j
j
−1
k
−k j
j −1
k
−k
limn→∞ Ψn (a)u = limn→∞ n
=
k=1 u au u = limn→∞ u n
k=1−j u au
j
limn→∞ u Ψn (a) in the ultraweak topology (note that the operation of left or right
multiplication by a unitary operator is continuous with respect to the ultraweak
topology). Moreover, ∀a ∈ M, x ∈ N, τ (Ψn (a)x) = τ (ax). Taking the limit n → ∞,
we get τ (Ψ(a)x) = τ (ax). Hence Ψ is a trace-preserving conditional expectation from
M onto N . Note that Ψn (x) = x, ∀x ∈ M 0 and hence so M 0 is in the multiplicative
domain of Ψ as n → ∞.
P
Therefore, for any ni=1 ai ⊗ xi ∈ M ⊗ M 0
n
n
n
X
X
X
Ψ(
ai x i ) =
EN (ai )xi = Φn (
ai ⊗ x i )
i=1
i=1
i=1
(9)
PERSONAL NOTES IN OPERATOR ALGEBRAS
27
For any compact operator a ∈ K(H), k ∈ N, uk au−k → 0 ultraweakly (or in WOT)
( Since u−k → 0 (WOT) and the fact that a is weak to strong continuous, hence for
any vectors ξ, η ∈ H, |hau−k ξ, u−k ηi| ≤ kau−k ξkkηk. . . ). By Cesaro’s mean theorem,
Ψ(a) = 0. Hence K(H) ⊂ Ker(Ψ). Then there is a unital completely positive map
Ψ̄ : B(H)/K(H) → B(H) such that Ψ = Ψ̄π (Note that Ψ is a unital completely
positive map from B(H) onto A0 and the above fact is the first isomorphism theorem
for the class of ucp maps and C*-algebras category). Thus ΦN = Ψ̄µ. (AO) condition
implies ΦN is continuous.
A consequence of this theorem is that L(G) is solid if G is hyperbolic or G is
a discrete subgroup of a connected simple Lie group of rank one. The following
interesting theorem tells us that there are infinitely many prime II1 factors with
property (T), i,e L(Γm ) for lattices Γm in Sp(1, m) ([5]) of Cowling-Haagerup.
Theorem 2.26. A subfactor of a solid factor is again solid and a solid factor is
prime and non-Γ if it is noninjective.
The proof of this theorem is merely a simple application of the following lemma
by Popa:
Lemma 2.27. Assume that the type II1 factor M with separable predual contains a
noninjective von Neumann subalgebra N0 such that N00 ∩M ω is a diffuse von Neumann
algebra. Then there exists a noninjective von Neumann subalgebra N1 ⊂ M such that
N10 ∩ M is diffuse.
Proof. If N is a subfactor of a solid factor M then consider any diffuse von Neumann
subalgebra A of N , then the relative commutant of A in N is equal to the intersection
of the relative commutant of A in M with N , or to be the image of the conditional
expectation EN of A0 ∩ M , which is injective. This shows solidity of N . Moreover,
if M is noninjective, M must be a II1 factor since M is a finite factor. Then if M
has property Γ, by a well-known result of Connes, M 0 ∩ M ω is diffuse. By Popa’s
lemma, there exists a noninjective subalgebra N ⊂ M such that A = N 0 ∩ M is
diffuse. Since M is solid and N ⊂ A0 ∩ M , this implies N is injective (contradiction!).
¯ 2 and
So M does not have property Γ. For the primeness result, write M = M1 ⊗M
assume M1 , M2 are infinite dimensional. By Tomita’s commutant theorem, Mi are
II1 -factors, for i = 1, 2. Since M1 ⊗ C, C ⊗ M2 are diffuse subalgebras of M , solidity
of M implies M1 , M2 is injective. Injectivity is preserved under taking von Neumann
algebra tensor product. This prove primeness of M .
¯ ∞ [0, 1]) ∗ LF∞ are nonCorollary 2.28. The following factors LF∞ and (LF∞ ⊗L
isomorphic.
¯ ∞ [0, 1]) ∗ LF∞ is solid then (LF∞ ⊗L
¯ ∞ [0, 1]) is a
Proof. Indeed, suppose (LF∞ ⊗L
solid by the above theorem. Since it is obviously noninjective, it follows by the above
theorem again that it is prime, which is a contradiction.
Definition 2.29. A II1 factor M is called semisolid if for every type II1 von Neumann subalgebra N , the relative commutant N 0 ∩ M is injective. M is semiexact if
it contains an ultraweakly dense exact C*-algebra.
28
MINH KHA
It is clear that solid implies semisolid and semisolid implies primeness for a noninjective type II1 factor by the proof of Theorem (2.26).
A Kurosh-type theorem of Ozawa was proved in [?Oz2]
Theorem 2.30. Every free product of semiexact type II1 factors is prime.
2.5. The Fuglede-Kadison determinant and the Brown spectral measure
in finite factors, and a brief survey on results of Haagerup-Schultz on the
hyperinvariant subspace problem in a II1 factor. A remarkable result which
is found in Haagerup and Schultz’s paper:
Theorem 2.31. Let M be a II1 factor with a faithful normal trace τ . Let T ∈ M .
−1
Then there exists a sequence Ak ∈ Minv such that kAk T A−1
k k ≤ kT k and Ak T Ak
converges in ∗-distribution to a normal operator N in the ultrapower M U such that
the Brown measure µT is equal to µN .
Roughly speaking, after passing to the ultrapower of M , the similarity orbit of T
in M would contain a normal operator N whose Brown measure is the same as T’s.
2.6. Some notes on Kaplansky’s conjectures. It is well-known in linear algebra
that any left (or right) - invertible matrix is both-sided invertible. We can ask the
same question for a von Neumann algebra M with a faithful tracial state τ . Here is
a proof of Burger and Valette for this general fact:
Theorem 2.32. Suppose that a, b ∈ M such that ab = 1. Then ba = 1. In other
words, M is a directly finite algebra.
Proof. If ab = 1 then baba = b1a = ba, or ba is an idempotent. Put e = 1 − ba
then e ∈ M is an idempotent such that τ (e) = 1 − τ (ab) = 0. We claim that e = 0
and hence ba = 1. If e is self-adjoint then this is obvious by the faithfulness of τ .
Otherwise, we use a trick to produce a projection f whose trace is the same as of e.
Let z = 1 + (e∗ − e)∗ (e∗ − e) then z ∈ M is invertible and moreover, ze = ez = ee∗ e.
Hence, z −1 ∈ {e}00 . Let f = ee∗ z −1 then f ∗ = f = f 2 . Moreover, since e = ee∗ ez −1 ,
τ (e) = τ (ee∗ ez −1 ) = τ (ee∗ z −1 e) = τ (ee∗ z −1 ) = τ (f ). Thus, f = 0 and so e = 0. Reference: [4].
2.7. Miscellaneous results on specific groups. The purpose of this subsection
is to collect some random results on group theory , especially on free product with
amalgamation of discrete groups.
The motivation of this below lemma comes from the Haagerup type condition on
a filtration of C*-algebra as compact quantum metric spaces which we encoutered
before in the sense of Ozawa and Rieffel [13]. Note that in that paper, they prove
that any discrete group containing a copy of Z2 does not have Haagerup property. An
open question is that if a group satisfies the Haagerup type condition then whether
or not it must be hyperbolic. Let us denote the class H consisting of all discrete
groups which satisfy the Haagerup type condition. It is proved that this class is
preserved under taking subgroup and free product. In a natural sense, we could
make a conjecture for the stability of H under taking the amalgamated free product
over a finite subgroup. The lemma might shed some light on this problem
PERSONAL NOTES IN OPERATOR ALGEBRAS
29
Lemma 2.33 (K, 2013). The group Z2 cannot be embedded inside an amalgamated
free product of two groups in the class H over a common finite subgroup or even a
torsion subgroup (i.e the order of every element in a torsion group is finite).
Proof. This is just an application of Theorem 4.5 in [11]. Suppose for contradiction.
Let x, y be two generators of the subgroup Z2 in G1 ∗H G2 where G1 , G2 ∈ H and H
is a tosion subgroup of G1 ∩ G2 . Then since xy = yx, there are three possibilities:
(a) Either x or y is conjugate to an element in H. This is impossible since the
order of x and y are infinite.
(b) If x and y are not conjugate to any element in H, but one of them is conjugate
to an element in one factor, let’s say it is G1 then the other is also conjugate to an
element in G1 too. Since conjugation is an equivalent relation, it implies that G1
contains a copy of Z2 , which is a contradiction with the assumption G1 ∈ H.
(c) If both x, y are not conjugate to any element in any factor Gi , then there exists
elements g, W ∈ G1 ∗H G2 , h, h0 ∈ H, integers j, k such that x = ghg −1 W j , y =
gh0 g −1 W k where ghg −1 , gh0 g −1 , W are commuting elements. Let m, n be orders of
h, h0 respectively. Then it is clear that xmkn = W jkmn = y njm . Such a relation cannot
be satisfied in Z2 . This gives us another contradiction.
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M.K., Department of Mathematics, Texas A&M University, College Station, TX
77843-3368, USA
E-mail address: kha@math.tamu.edu