The Dauns-Hofmann theorem and tensor
products of C*-algebras
Dave McConnell
Functional Analysis Seminar, Oxford, 22.10.13
C∗ -bundles
Definition
Let X be a locally compact Hausdorff space. A C*-bundle over X is a
triple (X , A, πx : A → Ax ), where A is a C∗ -algebra, πx : A → Ax
surjective ∗-homomorphisms, such that
T
1. x∈X ker πx = {0} (equivalently, kak = supx∈X kπx (a)k for all
a ∈ A),
2. for f ∈ C0 (X ) and a ∈ A, there is f · a ∈ A with
πx (f · a) = f (x)πx (a) for all x ∈ X .
`
I
may regard a ∈ A as a section â : X →
I
If for all a ∈ A the function x 7→ kπx (a)k ∈ C0 (X ), we say that
(X , A, πx : A → Ax ) is a continuous C∗ -bundle. Upper- and
lower-semicontinuos bundles are defined analogously.
Ax with â(x) = πx (a),
Examples of C∗ -bundles
1. By the Gelfand-Naimark theorem, any commutative C∗ -algebra is a
continuous C∗ -bundle; X = Max(A), Ax = C for all x ∈ X and πx
the evaluation maps.
2. Trivial bundles: for any C∗ -algebra B and locally compact Hausdorff
space X , A = C0 (X , B) (with pointwise operations and supremum
norm) is a continuous C∗ -bundle with constant fibre Ax = B.
3. Subtrivial bundles: with A trivial and A0 ⊆ A a C∗ -subalgebra of A
closed under the action of C0 (X ), we get a continuous C∗ -bundle
(X , A0 , σx : A0 → Bx ) with Bx ⊆ B for all x ∈ X .
Generalised Gelfand-Naimark Problem: given a non-commutative
C∗ -algebra A, can we choose a space X and ∗-homomorphisms
πx : A → Ax such that (X , A, πx : A → Ax ) is a C∗ -bundle.
Representation over Prim(A)
Let A be a C∗ -algebra and Prim(A) the set of kernels of irreducible
representations π : A → B(H). For P = ker π ∈ Prim(A) we identify the
quotient C∗ -algebra A/P ≡ π(A).
I
kak = sup{ka + Pk : P ∈ Prim(A)},
I
Hull-kernel topology has open sets of the form
U(I ) = {P ∈ Prim(A) : P 6⊇ I },
where I is an ideal of A,
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If Prim(A) is Hausdorff, then (Prim(A), A, πp : A → A/P) is a
continuous C∗ -bundle (Fell),
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In particular if A ≡ C0 (X ) is commutative then Prim(A) = X ,
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Prim(A) is non-Hausdorff in general (it is a locally compact
T0 -space),
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Dauns-Hofmann Theorem: represents any C∗ -algebra A as the
section algebra of a bundle over the complete regularisation of
Prim(A).
Glimm(A)
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Define an equivalence relation ≈ on Prim(A) via P ≈ Q iff
f (P) = f (Q) for all f ∈ C b (Prim(A)),
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As a set, let Glimm(A) = Prim(A)/ ≈, and let
ρA : Prim(A) → Glimm(A) be the quotient map,
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For each f ∈ C b (Prim(A)) define f ρ : Glimm(A) → C via
f ρ ([P]) = f (P), where [P] is the ≈-equivalence class of
P ∈ Prim(A),
Topology τcr on Glimm(A) induced by {f ρ : f ∈ C b (Prim(A))},
I
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For each p T
= [P] in Glimm(A), denote by Gp the closed two sided
ideal Gp = [P], the Glimm ideal of A corresponding to p,
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If A is unital then Glimm(A) = Max(Z (A)), and the Glimm ideals
are the ideals of A generated by the maximal ideals in Z (A).
The Dauns-Hofmann Theorem
In what follows we will assume that Glimm(A) locally compact.
Theorem (J. Dauns, K. Hofmann, 1968)
Let A be a C∗ -algebra. Then the triple (Glimm(A), A, πp : A → A/Gp ),
where
(i) Gp is the Glimm ideal of A corresponding to p ∈ Glimm(A), and
(ii) πp : A → A/Gp is the quotient ∗-homomorphism,
is an upper semicontinuous C∗ -bundle. The centre ZM(A) of the
multiplier algebra of A is ∗-isomorphic to C b (Glimm(A)), where
f ∈ C b (Glimm(A)) acts by pointwise multiplication,
πp (f · a) = f (p)πp (a) for all a ∈ A, p ∈ Glimm(A).
We will construct the Dauns-Hofmann representation of the tensor
product A ⊗ B of C∗ -algebras in terms of the corresponding
representations of A and B. Need to determine Glimm(A ⊗ B) in terms
of Glimm(A) and Glimm(B).
Continuous bundles over Glimm(A)
If Prim(A) is Hausdorff, then Prim(A) = Glimm(A) (as sets of ideals
and topologically), and the C*-bundle
(Glimm(A), A, πp : A → A/Gp ) = (Prim(A), A, πP : A → A/P)
is continuous.
More generally, Lee’s Theorem shows that
(Glimm(A), A, πp : A → A/Gp ) is a continuous C*-bundle if and only if
the canonical map ρA : Prim(A) → Glimm(A) is open (it is continuous
by construction).
Quasi-standard C*-algebras
Definition
An ideal I / A is said to be primal if, given ideals J1 , . . . , Jn of A with
J1 J2 . . . Jn = {0}, there is 1 ≤ i ≤ n with Ji ⊆ I .
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Primitive ideals are primal.
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A primal ideal is the kernel of the GNS representation associated
with a state φ of A which is a weak*-limit of factorial states
(Archbold & Batty).
Definition
A C*-algebra A is called quasi-standard if
(i) (Glimm(A), A, πp : A → A/Gp ) is a continuous C*-bundle, and
(ii) every Gp ∈ Glimm(A) is a primal ideal of A.
For separable A satisfying (i), (ii) is equivalent to the condition
(ii’) there is a dense subset D ⊆ Glimm(A) with Gp primitive for all
p ∈ D.
Examples
1. Let A be the C∗ -algebra of sequences x = (xn ) ⊂ M2 (C) such that
xn → x∞ ∈ M2 (C). Then Prim(A) is homeomorphic to N̂
(one-point compactification), where n ∈ N̂ is identified with the ideal
Pn = {x ∈ A : xn = 0}, 1 ≤ n ≤ ∞.
Thus the Dauns-Hofmann representation of A gives the trivial
bundle C (N̂, M2 (C)).
2. Let B = {x ∈ A : x∞ = diag(λ(x), µ(x))} ⊂ A. Then
Prim(B) = {Pn : n ∈ N} ∪ {ker(λ)} ∪ {ker(µ)}, where the Pn are
isolated and ker(λ) ≈ ker(µ). Thus
Glimm(B) = {Pn : n ∈ N} ∪ {ker(λ ⊕ µ)} ≡ N̂,
and the Dauns-Hofmann bundle (N̂, B, πn : B → Bn ) has fibres
M2 (C) if n ∈ N
Bn =
C ⊕ C if n = ∞,
and B is quasi-standard.
A discontinuous example
Take the C∗ -algebra C of sequences x = (xn ) ⊂ M2 (C) such that
x2n
→
diag(λ1 (x), λ2 (x))
x2n+1
→
diag(λ2 (x), λ3 (x)).
Then
Prim(C )
=
{Pn : n ∈ N} ∪ {ker(λi ) : i = 1, 2, 3}.
Glimm(C )
=
{Pn : n ∈ N} ∪ {ker(⊕3i=1 λi )} ≡ N̂
The bundle (N̂, C , πn : C → Cn ) has fibres Cn = M2 (C) for n ∈ N, and
C∞ = C3 , where π∞ (x) = (λ1 (x), λ2 (x), λ3 (x)).
Upper-semicontinuous, but not continuous: take x = (xn ) ∈ C with
x2n = diag(1, 0) and x2n+1 = 0.
Classes of Dauns-Hofmann representations
We have the following relations:
{ C∗ -algebras A with Prim(A) Hausdorff }
=
{ continuous C*-bundles over Prim(A)}
(
{ quasi-standard C*-algebras A}
(
{ continuous C*-bundles over Glimm(A)}
(
{ u.s.c. C*-bundles over Glimm(A)}
= { all C*-algebras }.
Question: are these classes closed under tensor products?
The minimal tensor product of C∗ -algebras
Let A and B be C∗ -algebras, A B their ∗-algebraic tensor product.
I
Let π : A → B(Hπ ), σ : B → B(Hσ ) be faithful representations,
then there is a ∗-homomorphism
π σ : A B → B(Hπ ) B(Hσ ) ⊆ B(Hπ ⊗ Hσ ) such that
(π σ)(a ⊗ b) = π(a) ⊗ σ(b),
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For t ∈ A B we define ktkα = k(π σ)(t)kB(Hπ ⊗Hσ ) ,
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k · kα is independent of faithful representations π, σ,
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Define A ⊗ B as the completion of A B with respect to k · kα ,
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For ∗-homomorphisms π : A → A0 , σ : B → B 0 we get a
∗-homomorphism π ⊗ σ : A ⊗ B → A0 ⊗ B 0 extending π σ,
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If π and σ are injective or surjective then π ⊗ σ has the same
property.
If k·kγ is another C∗ -norm on A B, A ⊗γ B will denote the completion
of A B in this norm.
Ideals of A ⊗ B
If I / A and J / B, let qI : A → A/I and qJ : B → B/J be the quotient
maps. Then qI qJ : A B → (A/I ) (B/J) has
ker(qI qJ ) = I B + A J,
which, by injectivity, has closure
I ⊗ B + A ⊗ J / A ⊗ B.
Extending qI qJ to qI ⊗ qJ : A ⊗ B → (A/I ) ⊗ (B/J) gives a closed
two-sided ideal
ker(qI ⊗ qJ ) / A ⊗ B.
Clearly
ker(qI ⊗ qJ ) ⊇ I ⊗ B + A ⊗ J.
but this inclusion may be strict,
If ker(qI ⊗ qJ ) = I ⊗ B + A ⊗ J for all I / A, J / B, we say that A ⊗ B
satisfies Tomiyama’s property (F).
Tensor Products of C∗ -bundles
Let A = (X , A, πx : A → Ax ) and B = (Y , B, σy : B → By ) be
C∗ -bundles. We can form the ’fibrewise tensor product’ of A and B, to
get a C∗ -bundle
(X × Y , A ⊗ B, πx ⊗ σy : A ⊗ B → Ax ⊗ By ),
(1)
(note that A ⊗ B/(ker(πx ⊗ σy )) ≡ Ax ⊗ By ).
Or we may take as fibres the quotient C∗ -algebras
A⊗B
≡ Ax ⊗x,y By =: (A ⊗ B)(x,y ) ,
ker(πx ) ⊗ B + A ⊗ ker(σy )
and get a C∗ -bundle
(X × Y , A ⊗ B, θx,y : A ⊗ B → (A ⊗ B)x,y ).
(2)
If A and B are continuous bundles, then (1) is lower-semicontinuous and
(2) is upper-semicontinuous (Kirchberg and Wassermann). In general
continuity may fail.
Characterisation of Glimm(A ⊗ B)
Theorem
Let A and B be C∗ -algebras. Then the map
Glimm(A) × Glimm(B) → Glimm(A ⊗ B)
(Gp , Gq ) 7→ Gp ⊗ B + A ⊗ Gq
is an open bijection , which is a homeomorphism if
(i) A is σ-unital and Glimm(A) locally compact (in particular if A
unital),
(ii) The complete regularisation map ρA : Prim(A) → Glimm(A) is an
open map (iff Dauns-Hofmann representation of A continuous).
Remark: in general, the topology on Glimm(A ⊗ B) depends only on
Prim(A) × Prim(B). We will use Glimm(A) × Glimm(B) to denote this
space with this possibly stronger topology, so that the above map is a
homeomorphism.
Sectional representation of A ⊗ B
Theorem
The Dauns-Hofmann representation
(Glimm(A ⊗ B), A ⊗ B, θr : A ⊗ B → (A ⊗ B)/Gr ) of A ⊗ B may be
canonically identified with the upper-semicontinuous C∗ -bundle
Glimm(A) × Glimm(B), A ⊗ B, θp,q : A ⊗ B → (A ⊗ B)(p,q) ,
where the fibre algebras are given by
(A ⊗ B)(p,q) =
A⊗B
= (A/Gp ) ⊗(p,q) (B/Gq ),
Gp ⊗ B + A ⊗ Gq
identifying r ∈ Glimm(A ⊗ B) with (p, q) ∈ Glimm(A) × Glimm(B)
such that Gp ⊗ B + A ⊗ Gq = Gr .
Corollary
ZM(A ⊗ B) is canonically ∗-isomorphic to C b (Glimm(A) × Glimm(B)).
The fibrewise tensor product
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The norm k · k(p,q) is the minimal C∗ -norm on (A/Gp ) (B/Gq ) if
and only if
Gp ⊗ B + A ⊗ Gq = ker(πp ⊗ σq ),
(∗)
where πp : A → A/Gp and σq : B → B/Gq are the quotient maps.
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A ⊗ B may be constructed as the minimal fibrewise tensor product
of the Dauns-Hofmann bundles of A and B, i.e.
(Glimm(A)×Glimm(B), A⊗B, πp ⊗σq : A⊗B → (A/Gp )⊗(B/Gq ))
if and only if (∗) holds for all (p, q).
Theorem
Let A and B be C∗ -algebras such that (∗) holds for all pairs of Glimm
ideals of A and B. Then A ⊗ B is a continuous C∗ -bundle over
Glimm(A ⊗ B) if and only if A and B are continuous bundles over
Glimm(A) and Glimm(B) respectively.
The fibrewise tensor product ii.
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What can we say about the condition
Gp ⊗ B + A ⊗ Gq = ker(πp ⊗ σq )
(∗)
for all (p, q) ∈ Glimm(A) × Glimm(B)?
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Kaniuth showed that if A ⊗ B has property (F), then (∗) holds.
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This condition is not necessary: B(H) ⊗ B(H) does not satisfy
property (F), but Glimm(B(H)) = {0}, so clearly (∗) holds.
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If A and B are continuous C∗ -bundles over their Glimm spaces then,
for (p0 , q0 ) ∈ Glimm(A) × Glimm(B), we have
(p, q) 7→ k(πp ⊗ σq )(c)k is continuous at (p0 , q0 ) if and only if (∗)
holds at this point (Archbold).
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No counterexamples given.
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Using a result of Kirchberg, we can construct C∗ -algebras A and B
for which
(p, q) 7→ kc + (Gp ⊗ B + A ⊗ Gq )k
(1)
is continuous on Glimm(A) × Glimm(B) for all c ∈ A ⊗ B, but
there is a c 0 ∈ A ⊗ B with
(p, q) 7→ k(πp ⊗ πq )(c 0 )k
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discontinuous at some (p0 , q0 ) ∈ Glimm(A) × Glimm(B):
Q
We take A = B(H) and B = n≥1 Mn (C), then Glimm(A) = {0}
and Glimm(B) = βN, hence
Glimm(A ⊗ B) = {B(H) ⊗ Gq : q ∈ βN}.
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(2)
We get q0 ∈ βN\N with B(H) ⊗ Gq0 ( ker(id ⊗ σq0 ).
Then (2) is discontinuous at q0 for some c 0 ∈ A ⊗ B, but we can
show that (1) is continuous (in fact A ⊗ B is quasi-standard).
Thus the condition (∗) is not equivalent to continuity of the
Dauns-Hofmann representation.
Exact C∗ -algebras
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Let B be a C∗ -algebra and J an ideal of B. Then the sequence
0
/J
ι
/B
q
/ (B/J)
/0
(†)
is called a short exact sequence of C∗ -algebras.
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We say that a C∗ -algebra A is exact if the sequence
0
/ A⊗J
id⊗ι
/ A⊗B
/ A ⊗ (B/J)
id⊗q
/0
is exact for every such sequence (†), that is, A ⊗ J = ker(id ⊗ q).
I
Examples include all commutative C*-algebras, liminal C*-algebras,
nuclear C*-algebras, subalgebras and quotients of nuclear
C*-algebras.
Exact C*-algebras
For H separable, then B(H) is inexact; in fact the short sequence
0
/ B(H) ⊗ K (H)
/ B(H) ⊗ B(H)
B(H) ⊗ (B(H)/K (H))
/0
is not exact; ker(id ⊗ qK ) ) B(H) ⊗ K (H) (Wassermann).
Theorem (Archbold, Batty, Kirchberg, Wassermann)
A is exact ⇔ A ⊗ B has property (F) for all C∗ -algebras B.
Note that (∗) is equivalent to exactness of
0
/ Gp ⊗ B + A ⊗ Gq
/ A⊗B
(A/Gp ) ⊗ (B/Gq )
/0
Characterisation of exactness
Theorem
For a C∗ -algebra A, the following are equivalent:
(i) A is exact,
(ii) For every C∗ -algebra B and q ∈ Glimm(B), the sequence
0
/ A ⊗ Gq
id⊗ι
/ A⊗B
id⊗σq
/ A ⊗ (B/Gq )
/0
is exact, where σq : B → B/Gq is the quotient map, (i.e.
A ⊗ Gq = ker(id ⊗ σq )).
(iii) For every C∗ -algebra B and (p, q) ∈ Glimm(A) × Glimm(B), we
have
A ⊗ Gq + Gp ⊗ B = ker(πp ⊗ σq ),
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Thus for a C*-algebra A: A ⊗ B has (∗) for all B ⇔ A exact ⇔
A ⊗ B has (F) for all B.
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But given A and B, A ⊗ B has (∗) 6⇒ A ⊗ B has (F).
Exact C*-bundles
Corollary
Let A and B be C∗ -algebras, one of which is exact. Then
(i) The Dauns-Hofmann representation of A ⊗ B over Glimm(A ⊗ B) is
a continuous C∗ -bundle if and only if the same is true for both A
and B,
(ii) A ⊗ B is quasi-standard if and only if A and B are,
(iii) Prim(A ⊗ B) is Hausdorff if and only if Prim(A) and Prim(B) are.
We will show that this ’stability property’ characterises exactness.
Theorem (Kirchberg & Wassermann)
Let (X , A, πx : A → Ax ) be a continuous C∗ -bundle. TFAE:
(i) A is an exact C∗ -algebra,
(ii) For any continuous C∗ -bundle (Y , B, σy : B → By ), the C∗ -bundle
(Y , A ⊗ B, id ⊗ σy : A ⊗ B → A ⊗ By )
is continuous,
(iii) For any continuous C∗ -bundle (Y , B, σy : B → By ), the C∗ -bundle
(X × Y , A ⊗ B, πx ⊗ σy : A ⊗ B → Ax ⊗ By )
is continuous.
Remarks:
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Ax exact ∀ x ∈ X 6⇒ A exact,
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For (ii) or (iii) ⇒ (i), enough to take B separable, unital, and
Y = N̂.
With A inexact, the Kirchberg & Wassermann result gives a separable,
unital, continuous C∗ -bundle (N̂, B, σn : B → Bn ) such that
(N̂, A ⊗ B, id ⊗ πn : A ⊗ B → A ⊗ Bn )
is discontinuous at ∞.
Using this we can construct a continuous bundle (N̂, C , γn : C → Cn ),
with the property that
(N̂, A ⊗ C , θn : A ⊗ C → (A ⊗ C )/(A ⊗ ker(γn )))
is discontinuous at ∞.
By a result of Blanchard, we may assume that (N̂, C , γn : C → Cn ) has
simple fibres.
It follows that Prim(C ) = Glimm(C ), canonically homeomorphic to N̂.
Thus, for each inexact A there is a continuous bundle
(N̂, C , γn : C → Cn ), with Prim(C ) = N̂, such that
(N̂, A ⊗ C , θn : A ⊗ C → (A ⊗ C )/(A ⊗ ker(γn )))
is discontinuous at ∞.
Suppose now that the bundle (Glimm(A), A, πp : A → A/Gp ) is
continuous.
As a consequence of the above, we can show that the Dauns-Hofmann
representation of A ⊗ C , i.e. the bundle
A⊗C
Glimm(A ⊗ C ), A ⊗ C , θp,n : A ⊗ C →
,
A ⊗ Gn + Gp ⊗ C
is necessarily discontinuous.
Characterisation of exactness
Theorem
Let A be a C∗ -algebra.
(i) If (Glimm(A), A, πp : A → A/Gp ) is a continuous C∗ -bundle, then A
is exact ⇔ for all C∗ -algebras B with
(Glimm(B), B, σq : B → B/Gq ) continuous, the bundle
(Glimm(A ⊗ B), A ⊗ B, θr : (A ⊗ B)/Gr ) is continuous,
(ii) If A is quasi-standard, then A is exact ⇔ A ⊗ B is quasi standard for
all quasi-standard C∗ -algebras B,
(iii) If Prim(A) is Hausdorff, then A is exact ⇔ Prim(A ⊗ B) is
Hausdorff for all C∗ -algebras B with Prim(B) Hausdorff.
Tensor products of C*-bundles
Thus none of the classes
{ C∗ -algebras A with Prim(A) Hausdorff }
=
{ continuous C*-bundles over Prim(A)}
(
{ quasi-standard C*-algebras A}
(
{ continuous C*-bundles over Glimm(A)}
are closed under tensor products. In each case, their intersection with
the class of exact C*-algebras is the largest ⊗-closed subclass.
The maximal tensor product
For C*-algebras A and B, the maximal C*-norm on A B is defined as
ktkmax = sup{ktkγ : k·kγ a C*-norm on A B}
for t ∈ A B.
A C*-algebra A is said to be nuclear if k·kmax = k·kα on A B for all
C*-algebras B.
Theorem
Let A be a unital quasi-standard C*-algebra. Then A is nuclear
⇔ A ⊗max B is quasi-standard for all quasi-standard C*-algebras B.