UNIVERSITY OF OSLO
Department of Mathematics
Faculty of Mathematics and
Natural Sciences
Simplicity of
Cuntz-Pimsner
algebras
M.Sc. thesis
Mathematics
Amandip S. Sangha
May 2007
1
Contents
1 Introduction
4
2 Basic definitions and results
5
3 Morita equivalence and Takai duality
13
4 The ideal structure of A ⋊X Z
20
5 Simplicity of OE
28
6 Cuntz-Pimsner algebras of self-similar group actions
34
2
3
1
Introduction
Given a C*-correspondence (E, ǫ) over a C*-algebra A, one may construct a
certain C*-algebra OE known as the Cuntz-Pimsner algebra (cf. [7]). This
construction generalizes for instance the Cuntz algebras, Cuntz-Krieger algebras and crossed products over Z by a single automorphism. There are
various results pertaining to simplicity of Cuntz-Pimsner algebras. This
paper contains a detailed exposition of one of the more general results regarding simplicity, namely
[10, Theorem 3.9] For a full C*-correspondence (E, ǫ) over a unital C*algebra A with ǫ faithful, the Cuntz-Pimsner algebra OE is simple if and
only if E is minimal and nonperiodic.
In addition to giving a thorough development of the material which culminates in the above main result, we consider some recently introduced
examples of Cuntz-Pimsner algebras arising from self-similar group actions.
The paper is organized as follows. In section 2 we give the basic terminology
and results regarding C*-correspondences, the construction of the CuntzPimsner algebra and crossed products over Hilbert C*-bimodules. Section
3 discusses Morita equivalence between crossed products over bimodules,
granted Morita equivalence between the underlying C*-algebras. Here we
also address Takai duality, i.e. determining the dual of the crossed product.
The ideal structure of crossed products over bimodules is analyzed in section
4, where, due to the Takai duality result achieved in section 3, one is able to
introduce the Connes spectrum into the study of the ideal structure. We are
then able to prove the simplicity result for crossed products over equivalence
bimodules, i.e. characterizing simplicity in terms of certain properties of the
bimodule. The main result on simplicity of the Cuntz-Pimsner algebra OE is
then achieved in section 5 by passage to the simplicity result for the crossed
product previously handled. In section 6 we apply the simplicity result to
the Cuntz-Pimsner algebra OΦ , where Φ is a C*-correspondence encoding
the self-similar action of a group G on a sequence space, as introduced in
[2]. Whereas in [2], simplicity of this Cuntz-Pimsner algebra was handled
in a way similar to that of the Cuntz algebra case, we deduce simplicity by
showing minimality and nonperiodicity of the module, thus producing an
alternative proof.
This thesis was written January—May 2007 for the degree of M.Sc. in
mathematics. I express my sincerest gratitude to my thesis advisor Assoc.
Prof. Sergey Neshveyev, from whom I have learnt much mathematics, and
I thank him for all his time, help and supervision on this work.
Amandip Singh Sangha,
Oslo, May 2007.
4
2
Basic definitions and results
A C*-correspondence (E, ǫ) over a C*-algebra A is a (right) Hilbert Amodule E together with a *-homomorphism ǫ : A −→ L(E). The module E
is thus equipped with a left action by A, namely we define a · ξ = ǫ(a)ξ, for
a ∈ A, ξ ∈ E. We shall assume the left action to be nondegenerate.
For any C*-algebra A, the identity correspondence over A consists of A considered as a Hilbert A-module, i.e. hx, yiA = x∗ y for x, y ∈ A, x · a = xa
and a · x = ax being the usual algebra multiplications.
A C*-correspondence (E, ǫ) is said to be full if hE, EiA = A.
Definition 2.1. For C*-correspondences (E, ǫ) over A and (F, φ) over B,
a semicovariant homomorphism π : (E, ǫ) −→ (F, φ) is a pair of maps
π = (πA , πE ), where πA : A −→ B is a *-homomorphism, and πE : E −→ F
is linear, such that for all a, b ∈ A and ξ, η ∈ E,
πE (aξ) = πA (a)πE (ξ),
πE (ξb) = πE (ξ)πA (b),
hπE (ξ), πE (η)iB = πA (hξ, ηiA ).
(1)
(2)
We call a semicovariant homomorphism π = (πA , πE ) nondegenerate
when πA is nondegenerate.
For a given C*-correspondence (E, ǫ), we shall construct a certain C*-algebra
which will be universal with respect to semicovariant homomorphisms from
(E, ǫ) to identity correspondences, i.e. any such semicovariant homomorphism will factor through this C*-algebra.
Proposition 2.2. Let (E, ǫ) be a C*-correspondence over A. There exists
a C*-algebra A ⋊E N and a nondegenerate semicovariant homomorphism
i : (E, ǫ) −→ A ⋊E N such that for every semicovariant homomorphism
π : (E, ǫ) −→ B to any identity correspondence B, there exists a unique
*-homomorphism πA × πE : A ⋊E N −→ B with πA = (πA × πE ) ◦ iA and
πE = (πA × πE ) ◦ iE .
.................................................
(E, ǫ) ......................................π
B
....
...
....
....
...
....
...
....
....
...
....
....
...
....
...
....
....
...
....
...
....
....
...
.....
........
i
.
.......
...
..
..
...
..
..
..
...
..
....
.
...
..
πA × πE
A ⋊E N
Proof. Denote by C the universal *-algebra generated by
{Tξ , Ta :
ξ ∈ E, a ∈ A}
5
subject to the following relations:
a 7→ Ta
is to be a *-homomorphism,
(3)
ξ 7→ Tξ
is to be a linear map,
(4)
Taξb = Ta Tξ Tb
Tξ∗ Tη
= Thξ,ηiA
for a, b ∈ A,
ξ ∈ E,
(5)
for ξ, η ∈ E.
(6)
Define
||T || = sup{ρ(T )
ρ
:
ρ C*-seminorm on C },
T ∈ C.
The supremum is bounded, since for any C*-seminorm ρ on C, the assignment a 7→ ρ(Ta ) defines a C*-seminorm on A, whence ρ(Ta ) ≤ ||a||. Also
ρ(Tξ )2 = ρ(Tξ∗ Tξ ) = ρ(Thξ,ξiA ) ≤ ||hξ, ξiA || = ||ξ||2 . Denote by A ⋊E N the
completion of C with respect to this norm (after passing to the appropriate
quotient). Then define the semicovariant homomorphism
i : (E, ǫ) −→ A ⋊E N
iA (a) = Ta
iE (ξ) = Tξ .
For any given C*-algebra B, which is considered as an identity correspondence over itself, and semicovariant homomorphism
π = (πA , πE ) : (E, ǫ) −→ B,
we define the *-homomorphism πA × πE : A ⋊E N −→ B on the generators
of C,
πA × πE (Ta ) = πA (a),
πA × πE (Tξ ) = πE (ξ)
and extend it in the obvious way to the completion. It is evident from our
construction that πA × πE ◦ iA = πA and πA × πE ◦ iE = πE .
Remark 2.3. We identify A with iA (A) ⊆ A ⋊E N and E with iE (E) ⊆
A⋊E N. One also uses the faithful Fock representation to understand the C*algebra A⋊E N as operators on the one-sided Fock space F 1 (E) = ⊕n≥0 E ⊗n ,
where E ⊗n = E ⊗ǫ · · · ⊗ǫ E, E ⊗0 = A. The Fock representation is then
defined on elementary tensors as
λA (a)ξ1 ⊗ · · · ⊗ ξk = aξ1 ⊗ · · · ⊗ ξk ,
λE (ζ)ξ1 ⊗ · · · ⊗ ξk = ζ ⊗ ξ1 ⊗ · · · ⊗ ξk ,
for a ∈ A, ζ ∈ X, and their adjoint operators are λA (a)∗ = λA (a∗ ) and
λE (ζ)∗ ξ1 ⊗ · · · ⊗ ξk = hζ, ξ1 iA ξ2 ⊗ · · · ⊗ ξk .
6
We see that (λA , λE ) : (E, ǫ) −→ L(F 1 (E)) is a semicovariant homomorphism,
λE (aζb)ξ1 ⊗ · · · ⊗ ξk = aζb ⊗ ξ1 ⊗ · · · ⊗ ξk
= aζ ⊗ bξ1 ⊗ · · · ⊗ ξk
= λA (a)λE (ζ)λA (b)ξ1 ⊗ · · · ⊗ ξk .
hλE (η), λE (ζ)iL(F (X)) = λE (η)∗ λE (ζ),
λE (η)∗ λE (ζ)ξ1 ⊗ · · · ⊗ ξk = λE (η)∗ ζ ⊗ ξ1 ⊗ · · · ⊗ ξk
= hη, ζiA ξ1 ⊗ · · · ⊗ ξk
= λA (hη, ζiA )ξ1 ⊗ · · · ⊗ ξk ,
thus by Proposition 2.2 (λA , λE ) factors through A⋊E N, inducing the unique
*-homomorphism λA × λE : A ⋊E N −→ L(F 1 (E)) with λA = λA × λE ◦ iA
and λE = λA × λE ◦ iE .
Later we shall put to use the Fock representation on the two-sided Fock
space, and also establish faithfulness of the representation (Remark 2.13).
Given a semicovariant homomorphism π : (E, ǫ) −→ (F, φ) we can define
a *-homomorphism πK(E) : K(E) −→ K(F ) by
πK(E) (θξ,η ) = θπE (ξ),πE (η) .
Definition 2.4. For C*-correspondences (E, ǫ) and (F, φ) over a C*-algebra
A, a semicovariant homomorphism π = (πA , πE ) : (E, ǫ) −→ (F, φ) is called
a covariant homomorphism if
holds on ǫ−1 (K(E)).
φ ◦ πA = πK(E) ◦ ǫ
We then commence by constructing a universal C*-algebra for covariant
homomorphisms to identity correspondences.
Proposition 2.5. Let (E, ǫ) be a C*-correspondence over A. There exists
a C*-algebra OE and a nondegenerate covariant homomorphism
j : (E, ǫ) −→ OE
such that for every covariant homomorphism π : (E, ǫ) −→ B to any identity
correspondence B, there exists a unique *-homomorphism π̂ : OE −→ B with
πA = π̂ ◦ jA and πE = π̂ ◦ jE .
..............................................
(E, ǫ) ......................................π
B
...
....
...
....
....
...
....
....
...
....
...
....
....
...
....
...
....
....
...
....
....
...
....
... .
.......
j
.
......
...
.
..
..
...
.
...
.
...
.
...
.
...
.
π̂
OE
7
Proof. Continuing with the construction from the proof of Proposition 2.2,
we use the map iK(E) : K(E) −→ A ⋊E N
iK(E) (θξ,η ) = Tξ Tη∗ ,
and define J to be the ideal in A ⋊E N generated by
a ∈ ǫ−1 (K(E))}.
{iK(E) ◦ ǫ(a) − iA (a) :
The Cuntz-Pimsner algebra is OE = A ⋊E N/J. Denoting the quotient
map q : A ⋊E N −→ OE and putting Sξ = q(Tξ ), Sa = q(Ta ), gives the
canonical covariant homomorphism j = (jA , jE ) : (E, ǫ) −→ OE , jA = q ◦iA ,
jE = q ◦ iE , and furthermore jK(E) = q ◦ iK(E) . The generators in OE thus
satisfy the relations
Saξb = Sa Sξ Sb ,
Sξ∗ Sη
= Shξ,ηiA ,
for a, b ∈ A, ξ ∈ E,
for ξ, η ∈ E,
Sa = jA (a) = jK(E) (ǫ(a)),
for a ∈ ǫ−1 (K(E)).
In particular, the last relation says that for an element a ∈ A such that
ǫ(a) = θξ,η , then Sa = Sξ Sη∗ .
Given any C*-algebra B (identity correspondence) and covariant homomorphism π = (πA , πE ) : (E, ǫ) −→ B, we define π̂ : OE −→ B by
π̂(Sa ) = πA (a) and π̂(Sξ ) = πE (ξ). Commutativity of the diagram follows
immediately from the construction.
Example 2.6. Let H be a Hilbert space of dimension n ≥ 2. If we choose
some orthonormal basis {ξi }i , then OH is generated by elements {Sξi }i and
jC (C), and we see from the relations that
Sξ∗i Sξj = Shξi ,ξj i = Sδij 1
and for any ζ ∈ H we have
Sξi Sξ∗i Sζ = Sξi Shξi ,ζi = Sξi hξi ,ζi .
Any P
ζ ∈ H has an expansion with respect to the orthonormal basis, i.e.
ζ = i ξi hξi , ζi, thus we get
X
X
Sξi hξi ,ζi = SPi ξi hξi ,ζi = Sζ .
Sξi Sξ∗i Sζ =
i
i
P
∗
i Sξi Sξi
It follows that
is the unit, denote it I, and by putting Si = Sξi we
can express the above in the more aesthetic and familiar form
Si∗ Si = I,
n
X
i = 1, . . . , n.
Si Si∗ = I,
i=1
which are the Cuntz algebra relations, hence OH ∼
= On .
8
Example 2.7. The Cuntz-Krieger algebra OB for any given n × n matrix
B = (bij )i,j with entries bij ∈ {0, 1} and each column and row nonzero, is
defined as the C*-algebra generated by partial isometries T1 , . . . , Tn whose
support projections Qi = Ti∗ Ti and range projections Pi = Ti Ti∗ satisfy the
relations
Pi Pj = 0, for i 6= j, and Qi =
n
X
bij Pj , i = 1, . . . , n.
j=1
In order to realize the Cuntz-Krieger algebra using a C*-correspondence, we
consider a finite set S = {s1 , . . . , sn } and the commutative finite dimensional
C*-algebra A = C(S). It will suffice to work with the basis elements fi ∈
C(S), denoting fi = χsi . Let E be the (isomorphism class of) finitely
generated (right) Hilbert A-module, generated by basis vectors
{eij :
bij = 1}.
We define a left module map φ : A −→ End(E) and define the module
structure
φ(fk )eij = δki eij
eij fk = δkj eij
heij , ei′ j ′ iA = δii′ δjj ′ fj .
By putting ei =
module, and
P
j eij ,
we have that {ei }i still generates E as a right
hei , ej iA = δij
X
bik fk .
k
Moreover
φ(fk )eij = δki eij = θek ,ek (eij ) = ek hek , eij iA = ek δki fj = δki ekj ,
Sei , Qi = Si∗ Si
i.e. φ(fk ) = θek ,ek . Using the symbols Sei , and defining Si =P
and Pi = Si Si∗ , we get precisely Pi Pj = 0 for i 6= j and Qi = j bij Pj , hence
OE ∼
= OB .
Definition 2.8. A Hilbert C*-bimodule (X, λ) over A is a C*-correspondence
over A which in addition is equipped with a left sided inner product
A h·, ·i
: X × X −→ A
such that
A hξ, ηiζ
= ξhη, ζiA ,
for all ξ, η, ζ ∈ X.
9
We note that we may also express the above relation by λ(A hξ, ηi) = θξ,η .
A Hilbert C*-bimodule (X, λ) over A is called an equivalence bimodule if
A hX, Xi = hX, XiA = A.
Any identity correspondence A becomes a Hilbert C*-bimodule by defining
the canonical left sided inner product A hx, yi = xy ∗ .
Definition 2.9. For Hilbert C*-bimodules (X, λ) over A and (Y, µ) over B,
a semicovariant homomorphism π = (πA , πX ) : (X, λ) −→ (Y, µ) is called a
bimodule homomorphism if
B hπX (ξ), πX (η)i
= πA (A hξ, ηi),
for all ξ, η ∈ X.
Proposition 2.10. Let (X, λ) be a Hilbert C*-bimodule over A. There exists a C*-algebra A ⋊X Z and a nondegenerate bimodule homomorphism
i : (X, λ) −→ A ⋊X Z such that for every bimodule homomorphism
π : (X, λ) −→ B
to any identity correspondence B (with the canonical left sided inner product),
there exists a unique *-homomorphism
πA × πX : A ⋊X Z −→ B
with πA = (πA × πX ) ◦ iA and πX = (πA × πX ) ◦ iX .
................................................
(X, λ) ....................................π
B
....
....
....
...
....
....
...
....
...
....
....
...
....
...
....
....
...
....
....
...
....
...
....
......
......
i
..
.......
...
...
.
...
.
...
.
...
.
...
.
...
.
πA × πX
A ⋊X Z
Proof. We add another relation to the already existing ones in the proof of
Proposition 2.2;
Tξ Tη∗ = TA hξ,ηi
for ξ, η ∈ X.
(7)
Then define A ⋊X Z as the completion of this universal *-algebra with the
norm as before. The bimodule homomorphism
i = (iA , iX ) : (X, λ) −→ A ⋊X Z
is naturally defined by iA (a) = Ta and iX (ξ) = Tξ , and πA ×πX (Ta ) = πA (a),
πA × πX (Tξ ) = πX (ξ) defines the *-homomorphism πA × πX .
10
Example 2.11. Considering a C*-algebra A and a fixed automorphism α ∈
Aut(A), one has the C*-dynamical system (A, Z, α) and the associated
crossed product A ⋊α Z. We define a Hilbert C*-bimodule X = A, endowed
with the module operations
a · x = α(a)x,
x · a = xa,
for a ∈ A, x ∈ X,
and inner products
A hx, yi
= α−1 (xy ∗ ),
hx, yiA = x∗ y.
With this module structure, one often denotes the module by α X to emphasize the automorphism in question. We have A ⋊α Z ∼
= A ⋊X Z. To show
this, we assume for convenience that A is unital. Let u ∈ A ⋊α Z be the
unitary which implements α. The element iX (1) ∈ A ⋊X Z is a unitary and
satisfies
iX (1)iA (a) = iX (a), iA (α−1 (a))iX (1) = iX (a),
and so we define a homomorphism
A ⋊α Z −→ A ⋊X Z
a 7−→ iA (a),
u 7−→ iX (1)∗ .
The inverse of this homomorphism is induced by the bimodule homomorphism
π = (πA , πX ) : X −→ A ⋊α Z
πA = idA ,
πX (a) = u∗ a.
Remark 2.12. We shall need to expand on the notion of Fock space from
Remark 2.3. For a Hilbert C*-bimodule X over A, we formally define the
adjoint bimodule X ∗ = {ξ ∗ : ξ ∈ X}, endowed with module operations
b · ξ ∗ · a = (aξb)∗ ,
A hξ
∗
, η ∗ i = hξ, ηiA ,
a, b ∈ A, ξ ∈ X,
hξ ∗ , η ∗ iA = A hξ, ηi, ξ, η ∈ X.
We adopt the conventions X ⊗0 = A, X ⊗n = X ⊗A · · · ⊗A X (interior tensor
product, n times), and X ⊗−n = (X ∗ )⊗n . The Fock space F(X) of X is
defined as
∞
M
X ⊗n .
F(X) =
n=−∞
The Fock representation is then constructed as follows. For a ∈ A we define
the operator λA (a) on F(X) by
λA (a)(ξn )n = (aξn )n ,
i.e. componentwise A-left action.
11
For ξ ∈ X define the operator λX (ξ) on F(X) as: for η ∈ X ⊗n

n > 0,

 ξ⊗η

ξη
n = 0,
λX (ξ)η =
∗i
hξ,
η
n
= −1,


 A
∗
⊗−1 ⊗ X ⊗n+1 .
A hξ, η1 iηn+1 n < −1, η = η1 ⊗ ηn+1 ∈ X
For ξ ∈ X ∗ the definition of λX (ξ) differs slightly,

ξ⊗η
n < 0,



ξη
n = 0,
λX (ξ)η =
∗
hξ, η i
n = 1,


 A
⊗n−1 .
∗
A hξ, η1 iηn−1 n > 1, η = η1 ⊗ ηn−1 ∈ X ⊗ X
In this notation we get λX (ξ)∗ = λX (ξ ∗ ).
For every z ∈ S 1 , (iA , ziX ) : (X, A) −→ A ⋊X Z is a bimodule homomorphism, which by Proposition 2.10 extends to a *-homomorphism
γz : A ⋊X Z −→ A ⋊X Z satisfying
γz (iA (a)) = iA (a), and γz (iX (ξ)) = ziX (ξ).
Then γ : z 7→ γz is a strongly continuous group of automorphisms, and γ is
referred to as the dual circle action.
For z ∈ S 1 we have unitaries Uz ∈ L(F(X)) which implement the so-called
gauge action on F(X), namely
Uz (ξn ) = z n ξn ,
for ξn ∈ X ⊗n ,
satisfying the identities
Uz λA (a)Uz∗ = λA (a),
Uz λX (ξ)Uz∗ = zλX (ξ),
where a ∈ A, ξ ∈ X, for all z ∈ S 1 .
Remark 2.13. Faithfulness of the Fock representation may be reasoned as
follows: λA is evidently faithful, for λX we employ the standard faithful
conditional expectations
Φ(x) =
Z
S1
and
Φ : A ⋊X Z −→ A ⋊X Z
γz (x) dz,
thus Φ(A ⋊X Z) = iA (A),
Ψ : L(F(X)) −→ L(F(X))
Z
Uz T Uz∗ dz,
Ψ(T ) =
S1
where we integrate using normalized Lebesgue measure. For an element
a ∈ ker(λA × λX ) we get λA × λX (Φ(a∗ a)) = Ψ(λA × λX (a∗ a)) = 0, and
faithfulness of λA × λX on iA (A) (which by restriction is just λA ) implies
a = 0.
12
3
Morita equivalence and Takai duality
Working with two Hilbert C*-bimodules, X over a C*-algebra A, and Y over
a C*-algebra B, where the underlying C*-algebras A and B are assumed
to be Morita equivalent, we will see that the crossed product C*-algebras
A ⋊X Z and B ⋊Y Z will be Morita equivalent if we are supplied with a
certain unitary map. In particular we will achieve that any crossed product
over a bimodule will be Morita equivalent to a certain crossed product of the
algebra of (generalized) compact operators using a partial automorphism.
This will comprise a technical ingredient for the main result of section 5.
We will also address Takai duality. In a more general setting, considering
a C*-dynamical system (A, G, α) and the dual system (A ⋊α G, Ĝ, α̂), it is
natural to ask about the structure of (A ⋊α G) ⋊α̂ Ĝ, and the Takai duality theorem states that this is isomorphic to A ⊗ K(L2 (G)). We shall
obtain a result analogous to this, when considering the C*-dynamical system (A ⋊X Z, S 1 , γ) with crossed product (A ⋊X Z) ⋊γ S 1 .
In the following we shall mean by an A—B C*-correspondence a module
which is a left Hilbert A-module and right Hilbert B-module, but with
only one right sided B-valued inner product. We will first be concerned
with extending such an A—B C*-correspondence to an A ⋊X Z—B ⋊Y Z
C*-correspondence, so that we later may (by using equivalence bimodules
instead of merely C*-correspondences) implement the promised Morita equivalence between the crossed products. This is handled in the following pair
of results.
Lemma 3.1. Let X be a Hilbert C*-bimodule over A, and B a C*-algebra.
For every pair (E, v) where E is an A—B C*-correspondence and
v : X ⊗A E −→ E ⊗B B
an isometric bimodule map, E extends to an (A⋊X Z)—B C*-correspondence
denoted E × v.
Proof. We define E × v = E and shall equip it with an A ⋊X Z-left module
action. If we can define a bimodule homomorphism
(λA , λX ) : (A, X) −→ L(E),
then by Proposition 2.10 it will factor through A ⋊X Z and thus yield an
A ⋊X Z-left module action on E.
Since E ⊗B B ≈ E canonically, we may write v : X ⊗A E −→ E. Define
λA : A −→ L(E) and λX : X −→ L(E) by
λA (a)η = aη,
for a ∈ A, η ∈ E,
λX (ξ)η = v(ξ ⊗ η),
i.e. usual A-left action on E,
for ξ ∈ X, η ∈ E.
13
These maps satisfy
λX (aξb)η = v(aξb ⊗ η) = av(ξ ⊗ bη) = aλX (ξ)bη,
in other words λX (aξb) = λA (a)λX (ξ)λA (b), and
hλX (ξ2 )∗ λX (ξ1 )η, ζiA = hv(ξ1 ⊗ η), v(ξ2 ⊗ ζ)iA = hξ1 ⊗ η, ξ2 ⊗ ζiA
= hη, hξ1 , ξ2 iA ζiA ,
so λX (ξ1 )∗ λX (ξ2 )ζ = hξ1 , ξ2 iA ζ, i.e. λX (ξ1 )∗ λX (ξ2 ) = λA (hξ1 , ξ2 iA ).
For an element ζ = v(ζ1 ⊗ ζ2 ) we have
hλX (ξ)η, ζiA = hv(ξ ⊗ η), v(ζ1 ⊗ ζ2 )iA = hξ ⊗ η, ζ1 ⊗ ζ2 iA = hη, hξ, ζ1 iA ζ2 iA
which means that λX (ξ)∗ ζ = hξ, ζ1 iA ζ2 . Consequently we get
λX (ξ1 )λX (ξ2 )∗ ζ = λX (ξ1 )hξ2 , ζ1 iA ζ2 = v(ξ1 ⊗ hξ2 , ζ1 iA ζ2 )
= v(ξ1 hξ2 , ζ1 iA ⊗ ζ2 ) = v(A hξ1 , ξ2 iζ1 ⊗ ζ2 )
= A hξ1 , ξ2 iv(ζ1 ⊗ ζ2 ) = A hξ1 , ξ2 iζ,
i.e. λX (ξ1 )λX (ξ2 )∗ = λA (A hξ1 , ξ2 i), and the bimodule homomorphism properties are thereby verified.
Now we get from Proposition 2.10 the unique *-homomorphism
λA × λX : A ⋊X Z −→ L(E),
and we may now define the A ⋊X Z-left action on E by
w · ξ = λA × λX (w)ξ,
for w ∈ A ⋊X Z, ξ ∈ E.
Corollary 3.2. Let E be an A—B C*-correspondence with an isometry
v : X ⊗A E −→ E ⊗B Y . Then E extends to an A ⋊X Z—B ⋊Y Z C*correspondence which we denote ⋊(E, v).
Proof. It is evident that E ⊗B B ⋊Y Z is an A—B ⋊Y Z C*-correspondence,
so if we can produce an appropriate isometric bimodule map, then we may
apply the preceding lemma. The isometry we are looking for is indeed
v ⊗ idB⋊Y Z : X ⊗A (E ⊗B B ⋊Y Z) −→ (E ⊗B B ⋊Y Z) ⊗B⋊Y Z B ⋊Y Z,
where we implicitly understand Y as embedded in B ⋊Y Z and thus E ⊗B Y
as embedded in E ⊗B B ⋊Y Z. Now by Lemma 3.1, E ⊗B B ⋊Y Z extends to
become an A ⋊X Z—B ⋊Y Z C*-correspondence, and we denote it ⋊(E, v).
14
Now we can conclude the discussion on how to implement Morita equivalence between the crossed products A ⋊X Z and B ⋊Y Z, given Morita
equivalence between A and B.
Theorem 3.3. Let X be a Hilbert C*-bimodule over A, and Y a Hilbert
C*-bimodule over B. If A ∼M B and there exists a unitary
u : X ⊗A E −→ E ⊗B Y
where E is the A—B equivalence bimodule implementing the Morita equivalence between A and B, then A ⋊X Z ∼M B ⋊Y Z.
Proof. By Corollary 3.2 ⋊(E, u) becomes an A ⋊X Z—B ⋊Y Z equivalence
bimodule, and hence implements the Morita equivalence
A ⋊X Z ∼M B ⋊Y Z.
Let X be a Hilbert C*-bimodule over A, and define F0 = F(X)A hX, Xi,
F1 = F(X)hX, Xi A . We get the ideals I = K(F0 ) and J = K(F1 ) in
K(F(X)). We have the unitary
u : F0 ⊗A X −→ F1
n−1
n−1
1
(ξn )n∈Z ⊗ ξ 7→ (ξn−1
⊗ · · · ⊗ ξn−1
⊗ λX (ξn−1
)ξ)n∈Z
n−1
where λX (ξn−1
) refers to the Fock operator as defined in Remark 2.12, using
1 ⊗ · · · ⊗ ξ m ∈ X ⊗m . The mapping t 7→ t ⊗ 1 maps
the notation ξm = ξm
m
K(F0 ) onto K(F0 ⊗A X) since hF0 , F0 iA = A hX, Xi, and we thus get a
partial automorphism αX of K(F(X)),
αX : I −→ J
αX (t) = u(t ⊗ 1)u∗ .
The ideal I is a Hilbert C*-bimodule, sometimes denoted αX K(F(X)), over
K(F(X)), where we endow I with the module operations
a · ξ = aξ,
usual multiplication between a ∈ K(F(X)) and ξ ∈ I,
ξ · a = α−1
X (αX (ξ)a)
for a ∈ K(F(X)), ξ ∈ I,
and inner products
K(F (X)) hξ, ηi
= ξη ∗ ,
hξ, ηiK(F (X)) = αX (ξ ∗ η),
for ξ, η ∈ I.
Since F(X) is a full C*-correspondence over A, i.e. hF(X), F(X)i A = A,
it is evident that A ∼M K(F(X)), where F(X) implements the Morita
equivalence. Furthermore F(X)⊗A X ≈ F(X)⊗A A hX, Xi⊗A X ≈ F0 ⊗A X,
and I ⊗K(F (X)) F(X) ≈ F1 ≈ F0 ⊗A X. In other words we have a unitary
I ⊗K(F (X)) F(X) ≈ F(X) ⊗A X. Hence we get the following corollary to
Theorem 3.3.
15
Corollary 3.4. For X a Hilbert C*-bimodule over A, we have
A ⋊X Z ∼M K(F(X)) ⋊I Z.
Remark 3.5. We may also sometimes write K(F(X)) ⋊αX Z for the above
crossed product, emphasizing the partial automorphism.
Remark 3.6. In the case where X is an equivalence bimodule over A, so that
we have A hX, Xi = hX, XiA = A, we get F0 = F1 = F(X) and evidently I =
J = K(F(X)), hence αX becomes an automorphism, αX ∈ Aut(K(F(X))).
One also observes that, upon denoting by Pn the operator of projection onto
the n-th summand of F(X), i.e.
Pn (ξk )k∈Z = ξn ∈ X ⊗n ,
that αX shifts the projection Pn one step forward to Pn+1 . Indeed, writing αX : K(F(X)) −→ M (K(F(X))) = L(F(X)) with strictly continuous
extension
αX : L(F(X)) −→ L(F(X))
we then have, considering an elementary operator θξ,η , ξ,η ∈ F(X), ζ =
(ζj )j ∈ F(X), ζj = ζj1 ⊗ · · · ⊗ ζjj−1 ⊗ ζ0 ∈ X ⊗j , fixed ζ0 ∈ X,
αX (Pn )(ζj )j = u(Pn ⊗ 1)u∗ (ζj )j
= u(Pn (ζj1 ⊗ · · · ⊗ ζjj−1 )j ⊗ ζ0 )
1
2
n
= ζn+1
⊗ ζn+1
⊗ · · · ⊗ ζn+1
⊗ ζ0
= Pn+1 (ζj )j .
Hence, αX (Pn ) = Pn+1 . One also sees that α acts as the identity on
λA (A) ∪ λX (X), the image of the Fock representation.
Now for the duality result which will ultimately be used to introduce
the Connes spectrum to our setting. We recall the notion of the dual circle
action γ : S 1 −→ Aut(A ⋊X Z)
γz (iA (a)) = iA (a),
γz (iX (ξ)) = ziX (ξ),
for z ∈ S 1 .
In the following result (cf. Takai duality), we consider the C*-dynamical
system (A⋊X Z, S 1 , γ) and determine the dual system ((A⋊X Z)⋊γ S 1 , Z, γ̂).
Theorem 3.7. Let X be an equivalence bimodule over A. Then
(A ⋊X Z) ⋊γ S 1 ∼
= K(F(X)).
16
Proof. Consider the C*-dynamical system (A ⋊X Z, S 1 , γ). The Fock representation (λA , λX ) from Remark 2.3 induces the *-homomorphism
λA × λX : A ⋊X Z −→ L(F(X)).
Denote by λS 1 : S 1 −→ L(F(X)) the gauge action on the Fock module,
that is, λS 1 (z) = Uz where Uz (ξ n ) = z n ξ n , for ξ n ∈ X ⊗n . It follows that
(λA × λX , λS 1 ) is a covariant representation of the C*-dynamical system
(A ⋊X Z, S 1 , γ) on F(X), that is
λA × λX (γz (iA (a))) = λS 1 (z)λA × λX (iA (a))λS 1 (z)∗
λA × λX (γz (iX (η))) = λS 1 (z)λA × λX (iX (η))λS 1 (z)∗ .
Using canonical embeddings σ : A ⋊X Z ֒→ M ((A ⋊X Z) ⋊γ S 1 ) and
S 1 ֒→ U M ((A ⋊X Z) ⋊γ S 1 ), the latter embedding being
denoted z 7→ uz ,
R
we know from the general theory that elements σ(x) S 1 f (z)uz dz, for
f ∈ L1 (S 1 ), span a dense subspace of (A ⋊X Z) ⋊γ S 1 . It suffices to use
the functions f (z) = z n , n ∈ Z, and interpreting ξ m ∈ X ⊗n ⊆
X Z as
R A⋊
−n
m
1
embedded ξ ֒→ M ((A ⋊X Z) ⋊γ S ), we can then define pn = S 1 z uz dz,
and say that elements ξ m pn , for m,n ∈ Z, span a dense subspace of
(A ⋊X Z) ⋊γ S 1 .
Since (λA × λX , λS 1 ) was a covariant representation of (A ⋊X Z, S 1 , γ) we
take the integrated form of it
Ψ = λA × λX ⋊ λS 1 : (A ⋊X Z) ⋊γ S 1 −→ L(F(X)),
which acts on the generators by
m
m
Ψ(ξ pn ) = λA × λX (ξ )
Z
z
−n
Uz dz = T
ξm
Z
z −n Uz dz
where by Tξ m is meant the operator Tξ m η = ξ m ⊗ η, Ri.e. the usual operator in the Fock representation. Upon writing Pn = z −n Uz dz, we have
Ψ(ξ m pn ) = Tξ m Pn ∈ L(F(X)). We note that Pn is the projection onto the
n-th summand of F(X). Indeed,
Z
Z
X
n
z k hζk , ηk iA dz
hPn (ζk )k , (ηk )k iA = z hUz (ζk )k , (ηk )k iA dz = z n
k
= hζn , ηn iA ,
hence Pn (ζk )k = ζn , and so we canonically identify X ⊗n = Pn F(X).
m p ) are limits of compact operators.
We now show that the elements Ψ(ξ
n
P
Choose {ηiα }α ⊆ X ⊗n such that { i A hηiα , ηiα i}α is an approximate unit in
A. For ζ ∈ F(X) we now get
X
i
θξ m ηiα ,ηiα (ζ) =
X
ξ m ηiα hηiα , ζiA = ξ m
i
i
=T
ξm
X
Pn ζ,
17
α α
A hηi , ηi iPn ζ
→ ξ m Pn ζ
hence Ψ(ξ m pn ) = Tξ m Pn ∈ K(F(X)). Furthermore,
Ψ(ξ m (η n )∗ pn ) = Tξ m Tη∗n Pn = θξ m ,ηn ,
for ξ m ∈ X ⊗m , η n ∈ X ⊗n , m, n ∈ Z, which establishes the assertion that
Ψ((A ⋊X Z) ⋊γ S 1 ) = K(F(X)). We now use the faithful left regular representation π ′ : (A ⋊X Z) ⋊γ S 1 −→ L(L2 (S 1 , A ⋊X Z)) which is given by
(π ′ (x)f )(w) = γw (x)f (w),
(π ′ (uz )f )(w) = f (z
−1
w),
x ∈ A ⋊X Z
2
1
f ∈ L (S , A ⋊X Z), w ∈ S 1 , z ∈ S 1
It is more convenient to consider this representation on l2 (Z) ⊗ A ⋊X Z, so
to this end we use the unitary isomorphism (Plancherel theorem)
F : L2 (S 1 ) −→ l2 (Z).
∼ L2 (S 1 ) ⊗ A ⋊X Z, we may reformulate the represSince L2 (S 1 , A ⋊X Z) =
entation π ′ accordingly, and then put
π(ξ m ) = (F ⊗ id)π ′ (ξ m )(F ∗ ⊗ id),
π(uz ) = (F ⊗ id)π ′ (uz )(F ∗ ⊗ id),
thus getting the representation
π : (A ⋊X Z) ⋊γ S 1 −→ L(l2 (Z) ⊗ A ⋊X Z).
It follows that for the canonical orthonormal basis {en }n in l2 (Z), and any
x ∈ A ⋊X Z, we have F ∗ (en ) = g, where g(z) = z n . Thus for ξ m ∈ A ⋊X Z,
m ∈ Z, we have
π ′ (ξ m )(g ⊗ x)(w) = γw (ξ m )(g ⊗ x)(w) = wm (g ⊗ ξ m x)(w),
and since wm g(w) = wm+n , we have F (wm g)(w) = em+n , so
π(ξ m )(en ⊗ x) = em+n ⊗ ξ m x.
Similarly, π ′ (uz )(g⊗x)(w) = (g⊗x)(z −1 w), and since g(z −1 w) = z −n wn and
F maps the latter function to z −n en , we have π(uz )(en ⊗ x) = z −n en ⊗ x.
Hence our representation
π : (A ⋊X Z) ⋊γ S 1 −→ L(l2 (Z) ⊗ A ⋊X Z)
is given by
π(ξ m )(en ⊗ x) = em+n ⊗ ξ m x,
π(uz )(en ⊗ x) = z −n (en ⊗ x),
18
ξ m ∈ X ⊗m ,
z ∈ S 1.
Define a map V : F(X) −→ l2 (Z) ⊗ A ⋊X Z by V (ξ m ) = em ⊗ ξ m . We may
now define a homomorphism VK(F (X)) : K(F(X)) −→ L(l2 (Z) ⊗ A ⋊X Z)
by
VK(F (X)) (θξ,η ) = θV (ξ),V (η)
for ξ, η ∈ F(X).
We claim that VK(F (X)) ◦ Ψ = π. Indeed, for n,m ∈ Z, ζ = (zn ) ∈ l2 (Z) and
x ∈ A ⋊X Z, we get
!
X
θξ mηiα ,ηiα (ζ ⊗ x)
VK(F (X)) (Ψ(ξ m pn ))(ζ ⊗ x) = VK(F (X)) lim
α
= lim
α
= lim
α
= lim
α
X
θV (ξ m ηiα ),V (ηiα ) (ζ ⊗ x) = lim
α
i
X
X
i
θem+n ⊗ξ m ηiα ,en⊗ηiα (ζ ⊗ x)
i
em+n ⊗ ξ m ηiα hen ⊗ ηiα , ζ ⊗ xiA⋊X Z
i
X
em+n ⊗ ξ m ηiα (ηiα )∗ (zn x) = em+n ⊗ ξ m zn x
i
m
= π(ξ pn )(ζ ⊗ x).
The fact that π was faithful and VK(F (X)) ◦ Ψ = π implies that Ψ is
injective, thus establishing (A ⋊X Z) ⋊γ S 1 ∼
= K(F(X)).
Remark 3.8. We wish to identify the system ((A ⋊X Z) ⋊γ S 1 , Z, γ̂) with the
system (K(F(X)), Z, αX ). We have established the isomorphism
Ψ : (A ⋊X Z) ⋊γ S 1 −→ K(F(X)),
so what remains to show is that the isomorphism also intertwines the respective actions, i.e.
Ψ ◦ γ̂ = αX ◦ Ψ.
To this end, observe that
m
γˆ1 (ξ pn ) = ξ
m
Z
z n zuz dz = ξ m pn+1 ,
Ψ(ξ m pn+1 ) = Tξ m Pn+1 ,
and on the other hand
Ψ(ξ m pn ) = Tξ m Pn ,
αX (Tξ m Pn ) = Tξ m Pn+1 , since αX (Pn ) = Pn+1 by Remark 3.6.
Thus we conclude Ψ ◦ γ̂ = αX ◦ Ψ and identify
((A ⋊X Z) ⋊γ S 1 , Z, γ̂) ∼
= (K(F(X)), Z, αX ).
19
4
The ideal structure of A ⋊X Z
For an equivalence bimodule X over A, we shall see that the Connes spectrum provides important information about the relationship between ideals
in A and ideals in A ⋊X Z. We shall use this to establish a characterization
of simplicity of A ⋊X Z. In this paper we always consider closed two-sided
ideals (additional conditions will be stated explicitly).
For a Hilbert C*-bimodule X over a C*-algebra A, one may consider the
map
⋊ : (A, X) −→ A ⋊X Z
to be a functor on the category of Hilbert C*-bimodules. This formalism
can be convenient to describe the relationship between the ideal structure
of A and that of A ⋊X Z.
To a bimodule homomorphism j : (A, X) −→ (B, Y ), the functor ⋊ assigns
the *-homomorphism
⋊(j) : A ⋊X Z −→ B ⋊Y Z
⋊(j)(iA (a)) = iB (jA (a)),
⋊(j)(iX (ξ)) = iY (jX (ξ)).
Definition 4.1. A sequence of bimodule homomorphisms
0
..............................................
j
........................ (C, Z) ...............................................
(A, X) .................................... (B, Y ) ............π
0
is called exact if the sequences
0
......................................................
A
0
.......................................................
X
j
A
......................................................
jX
.......................................................
B
Y
π
B
......................................................
πY
.......................................................
C
.........................................................
0
Z
.........................................................
0
are exact.
Definition 4.2. An ideal I ⊆ A ⋊X Z is called gauge invariant if γz (I) ⊆ I
for all z ∈ S 1 .
Proposition 4.3. Given an exact sequence
0
...............................................
j
........................ (C, Z) ...............................................
(A, X) ..................................... (B, Y ) ............π
0
then ⋊(j)(A ⋊X Z) is a gauge invariant ideal in B ⋊Y Z.
Proof. The image ⋊(j)(A ⋊X Z) is generated by iB (jA (A)) ∪ iY (jX (X)),
hence gauge invariance follows. Furthermore, we have
iB (im jA ) ∪ iY (im jX ) = iB (ker πB ) ∪ iY (ker πY ),
from which one deduces that ⋊(j)(A ⋊X Z) indeed is an ideal.
20
Proposition 4.4. An exact sequence
0
...............................................
j
........................ (C, Z) ...............................................
(A, X) ..................................... (B, Y ) ............π
0
induces an exact sequence
0
...........................................
⋊(j)
⋊(π)
A ⋊X Z ................................................................... B ⋊Y Z ................................................................. C ⋊Z Z ........................................... 0
Proof. We may denote by γzA and γzB the respective dual circle actions on
A ⋊X Z and B ⋊Y Z, and by ΦA and ΦB the respective faithful conditional
expectations (cf. Remark 2.13). The sequence commutes with the respective
dual circle actions, hence ΦB ◦ ⋊(j) = ⋊(j) ◦ ΦA . Let a ∈ ker ⋊ (j). Then
0 = ΦB (⋊(j)(a∗ a)) = ⋊(j)(ΦA (a∗ a)), and since ⋊(j) clearly is injective on
ΦA (A ⋊X Z) = iA (A), it follows that a = 0.
For surjectivity of ⋊(π), notice that im ⋊ (π) ⊇ iC (πB (B)) ∪ iZ (πY (Y )) =
iC (C) ∪ iZ (Z), and the latter generates C ⋊Y Z. Considering ⋊(π) modulo
the image of ⋊(j), we express the quotient map
q : B ⋊X Z/im ⋊ (j) −→ C ⋊Z Z.
Since im ⋊ (j) is a gauge invariant ideal in B ⋊Y Z, the quotient comes
equipped with a faithful conditional expectation which we can denote Φq .
Let a ∈ ker q. Then 0 = ΦC (q(a∗ a)) = q(Φq (a∗ a)), but Φq (a∗ a) is an
element of iB (B)/jA (A), and
im ⋊ (j) ∩ iB (B) = iB (im jA ) = iB (ker πB )
implies that q clearly is faithful on Φq (B ⋊Y Z/im ⋊ (j)) = iB (B)/jA (A),
hence a = 0. Thus q is also injective and we conclude that ker ⋊ (π) =
im ⋊ (j).
Definition 4.5. For a Hilbert C*-bimodule X over a C*-algebra A, an ideal
J ⊆ A is called X-bi-invariant if A hXJ, Xi ⊆ J and hX, JXiA ⊆ J.
Next we observe that an X-bi-invariant ideal naturally gives rise to an
exact sequences of bimodule homomorphisms, and by applying the functor ⋊
we get a gauge invariant ideal in the crossed product. For an X-bi-invariant
ideal J ⊆ A, consider the exact sequences
0
0
.......................................................
id
...........................................
J
id
.......................................................
A
⊗ id
πA
...................................................
id
A/J .................................................... 0
⊗π
X ⊗A J ....................X...........................J.......................... X ⊗A A ..........................X........................A.............................. X ⊗A A/J .......................................... 0
where the latter sequence can be also be read
0
.......................................................
..
..
..
XJ .................................................. XA ..................................... X(A/J) ....................................... 0
21
by using the canonical unitary equivalences, and by convention writing
X(A/J) for X ⊗A A/J. By Proposition 4.4 we get the exact sequence
0
⋊(id)
.......................................
⋊(π)
J ⋊XJ Z .............................................................. A ⋊X Z ................................................... A/J ⋊X(A/J) Z .................................. 0
It follows from Proposition 4.3 that J ⋊XJ Z = ⋊(id)(J ⋊XJ Z) ⊆ A ⋊X Z
is a gauge invariant ideal. And we can also establish a converse: that any
gauge invariant ideal in A ⋊X Z is precisely of this form. The next result
formalizes this discussion.
Proposition 4.6. There is a bijection between X-bi-invariant ideals in A
and gauge invariant ideals in A ⋊X Z, by mapping
J 7−→ J ⋊XJ Z
for an X-bi-invariant ideal J ⊆ A.
Proof. Injectivity of the mapping is immediate by using the faithful conditional expectation Φ from Remark 2.13, which yields Φ(J ⋊XJ Z) = iA (J).
If I ⊆ A ⋊X Z is a gauge invariant ideal, put J = i−1
A (I). This is an Xbi-invariant ideal in A. Indeed, we know that A ⋊X Z extends the module
multiplications and inner products of (A, X), the latter being universally
embedded in the former, so that we have
iA : A hXJ, Xi 7−→ iX (X)iA (J)iX (X)∗ = iX (X)IiX (X)∗ ⊆ I
and
iA : hX, JXiA 7−→ iX (X)∗ iA (J)iX (X) = iX (X)∗ IiX (X) ⊆ I.
Hence iA (A hXJ, Xi) ⊆ I and iA (hX, JXiA ) ⊆ I, from which
A hXJ, Xi
−1
⊆ i−1
A (I) = J, and hX, JXiA ⊆ iA (I) = J
follow. We clearly have J ⋊XJ Z ⊆ I, thus we get the surjection
π : A/J ⊗X⊗A (A/J) Z −→ A ⋊X Z/I,
and denoting by Φ1 and Φ2 the respective faithful conditional expectations,
we get for any a ∈ ker π that 0 = Φ2 (π(a)) = π(Φ1 (a)), but Φ1 (a) ∈ A/J,
hence we must have a = 0, so π is faithful and thus I = J ⋊XJ Z.
Regarding the Connes spectrum (cf. [6]) we need merely to recall that
for a C*-dynamical system (A, G, α) the Connes spectrum Γ(α) is a certain
b Relevant results regarding the Connes
closed subgroup of the dual group G.
spectrum and the ideal structure of the crossed product will be quoted.
22
Definition 4.7. For an equivalence bimodule X over a C*-algebra A, we
define Γ(X) = Γ(αX ), i.e. the Connes spectrum associated to the C*dynamical system (K(F(X)), Z, αX ).
We shall need to cite the following result on the Connes spectrum of the
dual system of a C*-dynamical system.
Lemma 4.8. (8.11.7, [6])
Let (A, G, α) be a C*-dynamical system and (A ⋊α G, Γ, α̂) its dual system.
An element t ∈ G belongs to the Connes spectrum G(α̂) of the dual system
if and only if I ∩ αt (I) 6= {0} for every non-zero closed ideal I ⊆ A.
In our setting, this result is carried over as follows.
Lemma 4.9. Let X be an equivalence bimodule over A. For z ∈ S 1 , we
have z ∈ Γ(X) if and only if
I ∩ γz (I) 6= {0}
for every closed ideal I ⊆ A ⋊X Z.
Proof. Consider the C*-dynamical
system (A ⋊X Z, S 1 , γ) with dual system
1
∼
(A ⋊X Z) ⋊γ S , Z, αX = (K(F(X)), Z, αX ), by Theorem 3.7. The result
follows immediately from Lemma 4.8 cited above.
We shall also need to cite the following result.
Lemma 4.10. (Lemma 2.1, [4])
Let (A, G, α) be a C*-dynamical system and assume that for each closed
nonzero ideal I ⊆ A and each t ∈ G we have I ∩ γz (I) 6= {0}. There is
then for each closed nonzero ideal J of A and each compact subset E of G
a nonzero element x ∈ J such that αt (x) ∈ J for all t ∈ E.
The preceeding result allows us to characterize fullness of the Connes
spectrum in terms of ideals.
Theorem 4.11. Let X be an equivalence bimodule over A. The following
statements are equivalent:
(i)
every nonzero closed ideal in A ⋊X Z contains a nonzero gauge
invariant ideal,
(ii)
Γ(X) = S 1 .
Proof. (i) =⇒ (ii). Γ(X) ⊆ S 1 follows by definition, for the reverse inclusion: let z ∈ S 1 and assume I ⊆ A ⋊X Z is a closed ideal. Then by
assumption I contains a gauge invariant ideal J ⊆ I, i.e. γz (J) ⊆ J. Then
obviously I ∩ γz (I) ⊇ γz (J) 6= {0}, hence z ∈ Γ(X) by Lemma 4.9.
(ii) =⇒ (i). Γ(X) = S 1 means that for all z ∈ S 1 we have I ∩ γz (I) 6= {0}
23
by Lemma 4.9, thus Lemma 4.10 is applicable. For a closed nonzero ideal
J ⊆ A ⋊X Z, use E = S 1 in Lemma 4.10 and get x ∈ J such that γz (x) ∈ J
for all z ∈ S 1 . Denote by J0 the ideal generated by x and {γz (x)}z∈S 1 in J.
Then J0 is a gauge invariant ideal in J.
In the following cited result Sp(α) refers to the Arveson spectrum (a
certain closed subset of the dual of the group in question, cf. [6]).
Lemma 4.12. (Theorem 4.5, [5])
If (A, G, α) is a C*-dynamical system such that A is G-simple, Γ(α) is discrete and Sp(α)/Γ(α) is compact, then Γ(α)⊥ is precisely the subgroup of
elements t ∈ G such that αt = Ad u for some unitary G-fixpoint u ∈ M (A).
Before we start on our main result about simplicity of crossed products
over bimodules, we need to undertake a short technical discussion regarding
periodicity. Working with a unitary u : X ⊗n −→ A for some n, it turns out
that we would benefit from u satisfying the additional requirement
u(x0 ⊗ · · · ⊗ xn−1 )xn = x0 u(x1 ⊗ · · · ⊗ xn )
for all x0 , . . . , xn ∈ X. The following discussion, which culminates in Lemma
4.13, shows that this additional requirement is safe to adopt after passing
to n2 .
Suppose we have a unitary equivalence β1 : X ⊗n −→ A. Using β1 we then
get two unitary equivalences X ⊗n+1 −→ X depending on whether we apply
β1 to the first n or last n components, i.e.
x0 ⊗ x1 ⊗ · · · ⊗ xn 7−→ x0 β1 (x1 ⊗ · · · ⊗ xn ), or
x0 ⊗ x1 ⊗ · · · ⊗ xn 7−→ β1 (x0 ⊗ · · · ⊗ xn−1 )xn
for xi ∈ X, i = 0, . . . , n. We know there exists a unitary z1 ∈ Z(M (A))
such that
x0 β1 (x1 ⊗ · · · ⊗ xn ) = z1 β1 (x0 ⊗ · · · ⊗ xn−1 )xn .
(8)
Define β2 = z1 β1 : X ⊗n −→ A. Re-applying the above argument to β2 we
get a unitary z2 , and again setting β3 = z2 β2 , we continue in this manner
till we get βn .
For x1 , . . . , xn , y1 , . . . , yn ∈ X, we have
β1 (x1 ⊗ · · · ⊗ xn−1 ⊗ xn β1 (y1 ⊗ · · · ⊗ yn )) = β1 (x1 ⊗ · · · ⊗ xn )β1 (y1 ⊗ · · · ⊗ yn ).
On the other hand, by successively applying the relations as in (8), we get
24
β1 (x1 ⊗ · · · ⊗ xn−1 ⊗ xn β1 (y1 ⊗ · · · ⊗ yn ))
= β1 (x1 ⊗ · · · ⊗ xn−1 ⊗ z1 β1 (xn ⊗ y1 ⊗ · · · ⊗ yn−1 )yn )
= β1 (x1 ⊗ · · · ⊗ xn−2 ⊗ xn−1 ⊗ β2 (xn ⊗ y1 ⊗ · · · ⊗ yn−1 )yn )
= β1 (x1 ⊗ · · · ⊗ xn−1 β2 (xn ⊗ y1 ⊗ · · · ⊗ yn−1 ) ⊗ yn )
= β1 (x1 ⊗ · · · ⊗ xn−2 ⊗ z2 β2 (xn−1 ⊗ xn ⊗ y1 ⊗ · · · ⊗ yn−2 )yn−1 ⊗ yn ),
continuing in this manner we finally arrive at
= β1 (x1 βn (x2 ⊗ · · · ⊗ xn ⊗ y1 ) ⊗ y2 ⊗ · · · ⊗ yn−1 ⊗ yn )
= β1 (zn βn (x1 ⊗ x2 ⊗ · · · ⊗ xn )y1 ⊗ y2 ⊗ · · · ⊗ yn )
= β1 (zn · · · z1 β1 (x1 ⊗ · · · ⊗ xn )y1 ⊗ · · · ⊗ yn )
= zn · · · z1 β1 (x1 ⊗ · · · ⊗ xn )β1 (y1 ⊗ · · · ⊗ yn )
and since, to begin with, this was equal to β1 (x1 ⊗ · · · ⊗ xn )β1 (y1 ⊗ · · · ⊗ yn ),
we conclude that zn · · · z1 = 1.
2
Lemma 4.13. Defining β : X ⊗n −→ A by
β(x1 ⊗ · · · ⊗ xn2 ) = β1 (x1 ⊗ · · · ⊗ xn ) · · · βn (xn2 −n+1 ⊗ · · · ⊗ xn2 )
we have that x0 β(x1 ⊗ · · · ⊗ xn2 ) = β(x0 ⊗ · · · ⊗ xn2 −1 )xn2 .
Proof. This is a matter of computation:
x0 β(x1 ⊗ · · · ⊗ xn2 ) = x0 β1 (x1 ⊗ · · · ⊗ xn ) · · · βn (xn2 −n+1 ⊗ · · · ⊗ xn2 )
= z1 β1 (x0 ⊗ x1 ⊗ · · · ⊗ xn−1 )xn β2 (xn+1 ⊗ · · · ⊗ x2n )
· · · βn (xn2 −n+1 ⊗ · · · ⊗ xn2 )
= z1 β1 (x0 ⊗ · · · ⊗ xn−1 )z2 β2 (xn ⊗ xn+1 ⊗ · · · ⊗ x2n−1 )
· · · x2n β3 (x2n+1 ⊗ · · · ⊗ x3n ) · · · βn (xn2 −n+1 ⊗ · · · ⊗ xn2 )
continuing in this way, we finally arrive at
= z1 β1 (x0 ⊗ · · · ⊗ xn−1 )z2 β2 (xn ⊗ · · · ⊗ x2n−1 )z3 β3 (x2n ⊗ · · · ⊗ x3n−1 )z4
· · · zn βn (xn2 −n ⊗ · · · ⊗ xn2 −1 )xn2
= z1 z2 · · · zn β1 (x0 ⊗ · · · ⊗ xn−1 )β2 (xn ⊗ · · · ⊗ x2n−1 )β3 (x2n ⊗ · · · ⊗ x3n−1 )
· · · βn (xn2 −n ⊗ · · · ⊗ xn2 −1 )xn2
= β(x0 ⊗ · · · ⊗ xn2 −1 )xn2 .
Definition 4.14. We call a Hilbert C*-bimodule X nonperiodic if X ⊗n ≈ A
implies n = 0.
25
Example 4.15. Continuing the situation of Example 2.11, we elaborate on
what nonexistence of bi-invariant ideals and nonperiodicity amount to for
the bimodule X = α X of that example. If J ⊆ A were an X-bi-invariant
ideal, then
A hxa, yi
= α−1 (xay ∗ ) ∈ J,
hx, ayiA = x∗ α(a)y ∈ J,
for all x, y ∈ X, a ∈ J,
for all x, y ∈ X, a ∈ J,
which is equivalent to α(J) = J. Regarding nonperiodicity, we first note
that we may identify
X ⊗m −→ αm X
x1 ⊗ · · · ⊗ xm 7−→ αm−1 (x1 )αm−2 (x2 ) · · · α(xm−1 )xm .
Now if X were to be periodic, say by a unitary equivalence
U ′ : X ⊗n −→ A
then in other words we would have a unitary
U : αn X −→ A.
In the bimodule αn X the left module multiplication is a · y = αn (a)y, for
a ∈ A and y ∈ αn X. Since U is a unitary bimodule map it satisfies U (a·y) =
aU (y), which yields
U (αn (a)y) = aU (y)
αn (a)y = U ∗ aU (y)
i.e. αn (a) = U ∗ aU , so αn is an inner automorphism. Hence, in this situation
we have
A has no X-bi-invariant ideals if and only if A has no α-invariant ideals.
X is nonperiodic if and only if αn is not inner for any n.
We now give a characterization of simplicity of a crossed product over
an equivalence bimodule.
Theorem 4.16. Let X be an equivalence bimodule over A. The crossed
product A ⋊X Z is simple if and only if A contains no X-bi-invariant ideals
and X is nonperiodic.
Proof. Assume that A contains no X-bi-invariant ideals and X is nonperiodic. By bijection there are no gauge invariant ideals in A⋊X Z. By Theorem
4.11 it will be sufficient to show Γ(X) = S 1 to establish simplicity of A⋊X Z.
Assume for contradiction that Γ(X) 6= S 1 , so Γ(X) must a finite subgroup
26
of S 1 . Considering the C*-dynamical system (A ⋊X Z) ⋊γ S 1 , Z, αX we
thus have Γ(X) = Γ(αX ) discrete, and Sp(αX ) being a closed subset of S 1 ,
is compact, hence Sp(αX )/Γ(αX ) is also compact. By Lemma 4.12 we then
have n ∈ Γ(αX )⊥ ⊆ Z such that αnX = Ad u, for some u ∈ U L(F(X)). Recall from Remark 3.6 that αX (Pm ) = Pm+1 and moreover αkX (Pm ) = Pm+k .
It now follows that uP0 u∗ = αnX (P0 ) = Pn , and applying this operator to an
element (ζm )m∈Z ∈ F(X), we get
αnX (P0 )(ζm )m = uP0 u∗ (ζm )m = Pn (ζm )m = ζn ∈ X ⊗n .
This yields a unitary equivalence A ≈ X ⊗n by mapping
P0 u∗ (ζm )m 7−→ uP0 u∗ (ζm )m = ζn ,
in contradiction to the assumption of nonperiodicity, hence we must have
Γ(X) = S 1 and thus A ⋊X Z must be simple.
Now assume that A ⋊X Z is simple. Since in particular there are no gauge
invariant ideals in A ⋊X Z, then by bijection there are no X-bi-invariant
ideals in A. Assume to the contrary that X is periodic, i.e. there is a unitary
u : X ⊗n ≈ A for some n 6= 0. Then we define a bimodule homomorphism
π = (πA , πX ) : (A, X) −→ L(A ⊕ X ⊕ X ⊗2 ⊕ . . . ⊕ X ⊗n−1 ) by
πX (ξ) = Tξ ,
πA (a) = Ta ,
for ξ ∈ X, a ∈ A,
as the usual representation on the Fock space, i.e. Tξ (η) = ξ ⊗ η and
Ta (η) = aη, but where we understand summands X ⊗n+k (in the Fock space)
as shifted back to X ⊗k for each k using the unitary u. Take an element
n−1
X
ηi ∈ A ⊕ X ⊕ X ⊗2 ⊕ . . . ⊕ X ⊗n−1 ,
ηi ∈ X ⊗i .
i=0
Let ξ1 , . . . , ξn be P
elements of X, and compute the composition Tξ1 Tξ2 · · · Tξn
n−1
on the element
i=0 ηi . Dealing with each term ηi one at a time, and
successively applying the unitary u whenever we reach n-tensors, we see
that
Tξ1 Tξ2 · · · Tξn (η0 ) = u(ξ1 ⊗ · · · ⊗ ξn η0 ) ∈ A
Tξ1 Tξ2 · · · Tξn (η1 ) = ξ1 u(ξ2 ⊗ · · · ⊗ ξn ⊗ η1 ) ∈ X
..
.
Tξ1 Tξ2 · · · Tξn (ηn−1 ) = ξ1 ⊗ · · · ⊗ ξn−1 u(ξn ⊗ ηn−1 ) ∈ X ⊗n−1 .
We may assume that the unitary u satisfies the extra condition as in Lemma
27
4.13 (after passing to n2 if need be), hence we get
n−1
X
Tξ1 Tξ2 · · · Tξn (
ηi )
i=0
= u(ξ1 ⊗ · · · ⊗ ξn η0 ) + ξ1 u(ξ2 ⊗ · · · ⊗ ξn ⊗ η1 )+
. . . + ξ1 ⊗ ξ2 u(ξ3 ⊗ · · · ⊗ ξn ⊗ η2 )
= u(ξ1 ⊗ · · · ⊗ ξn )η0 + u(ξ1 ⊗ · · · ⊗ ξn )η1 + . . . + u(ξ1 ⊗ · · · ⊗ ξn )ηn−1 .
This shows that Tξ1 · · · Tξn = Tu(ξ1 ⊗···⊗ξn ) , and setting
s = iX (ξ1 ) · · · iX (ξn ) − iA (u(ξ1 ⊗ · · · ⊗ ξn ))
we get s ∈ ker(πA × πX ). But on the other hand, s 6= 0 in A ⋊X Z, which
contradicts the assumed simplicity of A ⋊X Z. Thus we must have n = 0,
i.e. X must be nonperiodic.
5
Simplicity of OE
The initial idea of the proof of the simplicity criterion for OE is to identify
OE with a certain crossed product AE ⋊XE Z over a bimodule. One will
also need to take care of the properties of minimality and nonperiodicity
under this identification. We handle these issues (in brief) in the following
ancillary results, before reaching the main result in Theorem 5.7.
(n)
For each n ∈ N, define AE = iE (E) · · · iE (E)iE (E)∗ · · · iE (E)∗ , where
we have n factors of iE (E) and of iE (E)∗ , a C*-subalgebra in A ⋊E N
isomorphic to K(E ⊗n ). Collecting all these subalgebras together we get
P
(0)
(n)
the C*-algebra AE = ∞
n=0 AE , where by convention AE = A. We see
that AE = (A ⋊E N)γ is the gauge invariant subalgebra in A ⋊E N. Let
q : A ⋊E N −→ OE denote the quotient mapping (cf. proof of Proposition 2.5), and define AE = q(AE ) ⊆ OE . We also define the module
XE = q(iE (E))AE , which becomes a Hilbert C*-bimodule over AE when
equipped with the usual algebra multiplications and inner products
hx, yiAE = x∗ y,
AE hx, yi
= xy ∗ .
Lemma 5.1. OE ∼
= AE ⋊XE Z.
Proof. By the definitions of AE and XE we have a homomorphism
AE ⋊XE Z −→ OE .
The inverse homomorphism is obtained by considering the maps
q ◦ iA : A −→ A ⋊E N −→ AE ,
q ◦ iE : E −→ A ⋊E N −→ XE ,
28
thus getting the homomorphism q̃ : A ⋊E N −→ AE ⋊XE Z. Since clearly
(ker q̃ ∩ AE ) ⊃ (ker q ∩ AE ) and ker q is generated by (ker q ∩ AE ), we
have ker q ⊂ ker q̃, hence we get induced the desired inverse homomorphism
OE −→ AE ⋊XE Z.
We define a map
πn :
n
X
(k)
AE −→ L(E ⊗n )
k=0
πn (a) = a|E ⊗n ,
where we understand A ⋊E N ⊂ L(F 1 (E)) by the Fock representation, so to
be precise one has πn (a) = λA × λX (a)|E ⊗n in the notation of Remark 2.3.
Proposition 5.2. If E is full and ǫ is faithful, then
n
X
(k)
ker q ∩ (
AE ) = ker πn .
k=0
Proof. We refer to [10] for a complete proof, and instead, here we assume
A to be unital and E toPbe finitely generated for convenience.
We can find
P
∗
{ξi }m
⊂
E
such
that
θ
=
1.
Then
we
get
1
−
T
T
i=1 P
i ξi ,ξi
i ξi ξi = P0 and
Pn+1 = i Tξi Pn Tξ∗i , hence Pn ∈ ker q for each n ∈ N. We see that
Pn iE (E) · · · iE (E)iE (E)∗ · · · iE (E)∗ Pn = K(E ⊗n ) ⊂ L(F 1 (E)),
and more specifically, for ξ1 , . . . , ξk , ζ1 , . . . , ζl ∈ E, that
Pk Tξ1 · · · Tζk Tζ∗l · · · Tζ∗1 Pl = θξ1 ⊗···⊗ξk ,ζ1 ⊗···⊗ζl .
Hence K(F 1 (E)) ⊂ ker q. On the other hand we know that ker q is generated by compact operators, thus ker q ⊂ K(F 1 (E)), and therefore we
Pn
(l)
conclude that ker q = K(F 1 (E)). Let a ∈
l=0 AE , then πn+k (a) =
πn (a) ⊗ 1 ∈ L(E ⊗n ⊗ E ⊗k ). As ǫ is faithful and E is full, we conclude
that
X (l)
ker πn+k ∩
AE = ker πn .
l≤n
If T ∈ K(⊕k≤N
E ⊗k )
then
||a + T || ≥ ||(a + T )|E ⊗N+1 || = ||πN +1 (a)|| = ||πn (a)||,
Therefore
n
X
(k)
ker q ∩ (
AE ) = ker πn .
k≤0
29
N ≥ n.
Definition 5.3. For a C*-correspondence (E, ǫ) over a C*-algebra A, we
call E minimal if A contains no ideals J with hE, JEiA ⊆ J. We call E
nonperiodic if E ⊗n ≈ A implies n = 0.
An ideal which satisfies the property mentioned in the above definition
is also sometimes called invariant or E-invariant.
Lemma 5.4. Let (E, ǫ) be a C*-correspondence over A, and F a full Hilbert
A-module. Then
K(F ) −→ L(F ⊗A E)
t 7−→ t ⊗ 1
maps K(F ) onto K(F ⊗A E) if and only if ǫ(A) = K(E).
Proof. Assume ǫ(A) = K(E). For z, w ∈ E, find a ∈ A such that ǫ(a) =
θz,w . Let x, y, u ∈ F , and v ∈ E. One computes that
θx⊗z,y⊗w (u ⊗ v) = (θxa,y ⊗ 1)(u ⊗ v)
thus we map θxa,y 7→ θxa,y ⊗ 1 = θx⊗z,y⊗w , and since the range of this map is
closed and K(F ⊗A E) is linearly generated by elements of the form θx⊗z,y⊗w ,
it follows that t 7→ t ⊗ 1 maps onto as claimed.
Conversely, assume that t 7→ t ⊗ 1 maps K(F ) onto K(F ⊗A E). We know
that a 7→ a ⊗ idF ⊗A E maps A onto K(F ∗ ⊗K(F ⊗A E) (F ⊗A E)). But
F ∗ ⊗K(F ⊗A E) (F ⊗A E) = F ∗ ⊗K(F ) (F ⊗A E) = (F ∗ ⊗K(F ) F ) ⊗A E
≈ A ⊗A E ≈ E,
from which the claim follows.
Lemma 5.5. Let (E, ǫ) be a full C*-correspondence over a unital C*-algebra
A, with ǫ faithful. Then XE ≈ AE implies E ≈ A.
Proof. Assume we have a unitary U : XE −→ AE . Putting α = U ◦ jE , we
have α(ξa) = α(ξ)jA (a) and α(ξ)∗ α(η) = jA (hξ, ηiA ) for all ξ,η ∈ E and
a ∈ A. Furthermore α(E)AE = AE . For any ξ, η, ζ ∈ E we have
α(ξ)α(η)∗ α(ζ) = α(ξhη, ζiA ) = U (jE (θξ,η (ζ)))
= U (jK(E) (θξ,η )jE (ζ)) = jK(E) (θξ,η )U (jE (ζ))
= jK(E) (θξ,η )α(ζ),
thus α(ξ)α(η)∗ = jK(E) (θξ,η ). Inductively we obtain
(n)
α(E) · · · α(E)α(E)∗ · · · α(E)∗ = AE ,
30
from which it follows that
(n)
(n+1)
α(E)AE α(E)∗ = AE
(n+1)
α(E)∗ AE
,
(n)
α(E) = AE .
(n)
Denote by J the image of ∪∞
n=1 AE ⊆ AE in AE . Then
α(E) = α(E)α(E)∗ α(E) = jK(E) (K(E))α(E) ⊆ J,
hence jA (A) = α(E)∗ α(E) ⊆ J, so J = AE . We also know that jK(E) (K(E))
= α(E)α(E)∗ contains an approxamite unit for AE since jK(E) is nonde(n)
(n+1)
for each n. We wish to show that this is an
generate. Thus AE ⊆ AE
equality. The ideal
hE,
Ei
is
dense
in A, so there
A
P
P exist ∗finite sets {ξi }i , {ηi }i
in E such that i hξi , ηi iA = 1, which implies i α(ξi ) α(ηi ) = 1. Choose n
P
(n)
such that there exist {yi }i ⊆ AE with ||yi − α(ηi )|| ≤ (6 ||α(ξj )||)−1 and
P
P
(n)
|| α(ξi )∗ yi || ≤ 2. Define z =
α(ξi )∗ yi , then for any a ∈ AE such that
(n−1)
and
||a|| ≤ 1, we have zaz ∗ ∈ AE
||zaz ∗ − a|| ≤ ||(z − 1)az ∗ || + ||a(z ∗ − 1)|| ≤ ||z − 1||(||z|| + 1)
X
X
= 3||
α(ξi )∗ yi − 1|| = 3||
α(ξi )∗ (yi − α(ηi ))||
X
1
≤3
||α(ξi )|| ||yi − α(ηi )|| ≤ .
2
But for a proper closed subspace F0 of a Banach space F we know that for
an arbitrary ǫ > 0 there exists an element in the unit ball of F with distance
(n)
(n−1)
cannot be a proper subspace of AE ,
to F0 greater than 1 − ǫ. Thus AE
and it follows that k 7→ k ⊗ idE maps K(E ⊗n−1 ) onto K(E ⊗n−1 ⊗A E).
(n+1)
(n)
for
Using Lemma 5.4 we conclude that ǫ(A) = K(E), thus AE = AE
all n. Therefore AE ∼
A,
and
hence
E
≈
X
≈
A
≈
A.
=
E
E
Let (E, ǫ) be a full C*-correspondence over A. For an ideal J ⊆ A, one
defines a set
E −1 (J) = {a ∈ A : hE, aEiA ⊆ J},
and says that the ideal J is saturated if E −1 (J) ⊆ J. For any proper
ideal J ⊆ A, we have that E −1 (J) is also a proper ideal. If J is an ideal
such that hE, JEiA ⊆ J, then J ⊆ E −1 (J). Maximal invariant ideals
are saturated; if J is a maximal invariant ideal, then E −1 (J) is invariant and contains J, hence must equal J. If the C*-algebra A is unital,
then the property of minimality is equivalent to nonexistence of nontrivial
saturated and invariant ideals. For Hilbert C*-bimodules, saturated invariant ideals are bi-invariant, since assuming hX, JXiA ⊆ J implies that
hX, A hXJ, XiXiA = hX, XJhX, XiA iA = hX, XiA JhX, XiA ⊆ J, thus also
31
A hXJ, Xi
⊆ J. We also remark that for a C*-correspondence (E, ǫ), minimality implies faithfulness of ǫ, since ker ǫ would have been an invariant
ideal.
Lemma 5.6. Let (E, ǫ) be a full C*-correspondence over a unital C*-algebra
A, with ǫ faithful. Then (E, ǫ) is minimal and nonperiodic if and only if
(XE , lE ) is minimal and nonperiodic.
Proof. Assume that (XE , lE ) is periodic, i.e. there is an n such that XE⊗n ≈
XE ⊗n ≈ AE . Then by Lemma 5.5 we have that E ⊗n ≈ A, i.e. E is periodic.
On the other hand, if E ⊗n ≈ A, then XE⊗n ≈ XE ⊗n ≈ AE .
Assume that (E, ǫ) is minimal. Let J ⊆ AE be an ideal such that
hXE , JXE iAE ⊆ J.
Since A is unital and due to the discussion of saturated ideals, we can assume
that J is XE -bi-invariant. The bimodule homomorphism
(qAE , qXE ) : (XE , λE ) −→ (XE , lE )
−1
(J) is XE -bi-invariant, and in particular that
gives that J˜ = qA
E
˜ ⊗n iA ⊆ J˜
hE ⊗n , JE
and
˜ E ⊗n i) ⊆ J˜
iK(E ⊗n) (K(E ⊗n ) hE ⊗n J,
for each n > 0. We have that iK(E ⊗n ) (K(E ⊗n )) ∩ J˜ 6= 0 if and only if
jK(E ⊗n ) (K(E ⊗n )) ∩ J 6= 0, moreover iK(E ⊗n) (K(E ⊗n )) ∩ J˜ 6= 0 implies that
˜ ⊗n iA ⊆ J˜∩A is nonzero. Minimality of (E, ǫ)
hE ⊗n , (iK(E ⊗n ) (K(E ⊗n ))∩ J)E
now implies that J˜ ∩ A = A, from which it follows that
iK(E ⊗n ) (K(E ⊗n )) ⊆ iK(E ⊗n ) (K(E ⊗n ) hE ⊗n (J˜ ∩ A), E ⊗n )i
⊆ iK(E ⊗n ) (K(E ⊗n )) ∩ J˜ ⊆ iK(E ⊗n ) (K(E ⊗n ))
since J˜ was XE -bi-invariant. This contradicts that J˜ was proper, hence
P
(k)
jK(E ⊗n ) (K(E ⊗n )) ∩ J = 0 for each n, i.e. J ∩ k≤n AE = 0 since it
is orthogonal to the essential ideal jK(E ⊗n ) (K(E ⊗n )). Since the span of
(n)
∪∞
n=1 AE was dense in AE , it follows that J = 0.
Conversely, let I ⊆ A be a nontrivial ideal such that hE, IEiA ⊆ I. One
may again assume that I is saturated. We get a nontrivial ideal
J = {x ∈ A ⋊E N :
hF 1 (E), λA × λX (x)F 1 (E)iA ⊆ I}
in A ⋊E N. This is because I ⊆ J and also J ∩ A ⊆ I. Then J maps to a
notrivial ideal in OE by faithfulness of jA (following from that of ǫ).
32
Theorem 5.7. Let (E, ǫ) be a full C*-correspondence over a unital C*algebra A, with ǫ faithful. Then OE is simple if and only if E is minimal
and nonperiodic.
Proof. Assuming OE ∼
= AE ⋊XE Z to be simple, it follows from the second
part of the proof of Theorem 4.16 that AE has no XE -bi-invariant ideals
and XE is nonperiodic regardless of XE being an equivalence bimodule. It
then follows from Lemma 5.6 that E is minimal and nonperiodic.
Conversely, assume that E is minimal and nonperiodic. Then by Lemma
5.6 (XE , lE ) is also minimal and nonperiodic. We have OE ∼
= AE ⋊XE Z,
but since (XE , lE ) may not be an equivalence bimodule in general, Theorem
4.16 is not applicable immediately. So in the following we assume (XE , lE )
to not be an equivalence bimodule. By Corollary 3.4 we have
AE ⋊XE Z ∼M B ⋊β Z,
where β : I −→ B is a partial automorphism, I = K(F(XE )AE hXE , XE i) is
an essential ideal in B = K(F(XE )). Compose the extension
β : M (I) −→ B
with the restriction map M (B) −→ M (I) to get an injective *-homomorphism
β̃ : M (B) −→ M (B), so B ֒→ β̃(B). Then {β̃ n (B)}n∈N is an increasing sequence of ideals
B ֒→ β̃(B) ֒→ β̃ 2 (B) ֒→ β̃ 3 (B) ֒→ · · ·
n
and we put C = ∪∞
n=1 β̃ (B). We get an automorpism γ = β̃|C ∈ Aut(C).
Denote by X the Hilbert C*-bimodule corresponding to β̃, i.e. X = B and
module multiplication and inner products are defined by
a · x · b = β̃(a)xb,
B hx, yi
= β̃ −1 (xy ∗ ),
hx, yiB = x∗ y,
a, b, x, y ∈ B.
Using the isomorphism β between I and B we thus understand B ⋊β Z (cf.
Corollary 3.4) as B ⋊X Z. Minimality of XE means there are no gauge invariant ideals in AE ⋊XE Z, and by Morita equivalence there can be no gauge
invariant ideals in B ⋊X Z, hence B can have no X-bi-invariant ideals. We
denote by (Y, γ) the Hilbert C*-bimodule corresponding to γ (with Y = C),
which then is an equivalence bimodule.
Simplicity of C ⋊Y Z = C ⋊γ Z is, by Theorem 4.16, equivalent to nonperiodicity of Y and nonexistence of Y -bi-invariant ideals in C. If Y were
periodic, say Y ⊗m ≈ C, then γ m (B) = B, hence we would have C = B so Y
would in fact be an equivalence bimodule, thus also (XE , lE ) to begin with,
contrary to our working assumption.
Assume that J ⊆ C is Y -bi-invariant ideal. Then J0 = B ∩ J is an ideal
in B with hX, J0 XiB = B ∗ β̃(J0 )B ⊆ B ∗ γ(J)B ⊆ J ∩ B. Likewise we have
33
= β̃ −1 (BJ0 B) ⊆ J ∩ B. Hence J0 = B since B had no X-biinvariant ideals, thus J = C.
We now show B⋊X Z ∼
= C⋊Y Z. Since B ֒→ C ⊆ M (B), and then in particular X ֒→ Y , the inclusion induces a *-homomorphism π : B⋊X Z −→ C⋊Y Z.
We also have a bimodule homomorphism (ψC , ψY ) : (Y, γ) −→ M (B ⋊X Z)
where
ψC is the restriction of M (B) ֒→ M (B ⋊X Z) to C,
B hXJ0 , Xi
ψY (y) = vψC (y), where v is the isometry s.t. β(b) = v ∗ bv.
Hence there exists a unique *-homomorphism ψ : C ⋊Y Z −→ M (B ⋊X Z),
and moreover ψ ◦ π = id, since π was an inclusion in one direction, and ψ is
the restriction in the opposite direction. Hence π is injective. Since ψ maps
π(B ⋊X Z) onto the ideal B ⋊X Z ⊆ M (B ⋊X Z), thus π(B ⋊X Z) is itself
an ideal, but by simplicity of C ⋊Y Z we must have π(B ⋊X Z) = C ⋊Y Z,
i.e. π is surjective.
In Example 2.6 the Cuntz algebra On was realized as OH using H = Cn .
The module H is of course minimal since C is already simple. Nonperiodicity
of H is also immediate since
H ⊗m = Cn ⊗ · · · ⊗ Cn ≈ C
is impossible for any m 6= 0 by taking the dimensions into account. Hence the
application of the simplicity result is trivial for the classical Cuntz algebra.
6
Cuntz-Pimsner algebras of self-similar group actions
Given a self-similar group action of a group G on a sequence space X ω ,
one associates a bimodule Φ over the group algebra CG, encoding the selfsimilarity. Considering a certain completion AΦ of CG, one has that Φ
becomes a C*-correspondence over AΦ . In this section we aim to apply the
simplicity result to the Cuntz-Pimsner algebra OΦ .
We begin by recalling the basic terminology and facts regarding self-similar
group actions, self-similar completions and self-similarity bimodules from
[2].
Let X be a finite set, and denote by X ω the set of all infinite sequences
(words) of the form x1 x2 x3 . . ., where xi ∈ X for each i. Equip X ω with
the direct product (Tikhonov) topology coming from the discrete sets X.
The basis of the topology then consists of cylindrical sets a1 a2 a3 . . . an X ω ,
ai ∈ X. One denotes by X ∗ the set of all finite words x1 . . . xn , together
34
with the empty word, thus X ∗ = ∪n≥0 X n . Given a sequence w ∈ X ω and a
finite word v, we understand vw as the concatenation of the two.
We shall be concerned with a countable group G acting faithfully on the
space X ω , writing g(w) for the action of g ∈ G on w ∈ X ω . The group
action will be a so-called self-similar group action, meaning that for every
g ∈ G and x ∈ X there exist h ∈ G and y ∈ X such that
g(xw) = yh(w),
for all w ∈ X ω .
One writes the last equation, referred to as the self-similarity condition,
formally as
g · x = y · h.
In the following we fix a self-similar and minimal group action of a group G
on X ω for a finite set X = {x1 , . . . , xd }. Denote by Φ the free (right) CGmodule, free basis being X. The module Φ has the CG-valued sesquilinear
form
+
*
X
X
X
=
a∗x bx .
x · ax ,
x · bx
x∈X
x∈X
CG
x∈X
We define a left module multiplication using the self similarity condition,
namely for any g ∈ G and x ∈ X, we define the left multiplication by
g · x = y · h, where h ∈ G and y ∈ X are such that g(xw) = yh(w) for all
w ∈ X ω . The multiplication extends by linearity to the whole module Φ,
giving a map from G to End(Φ). This map then extends to
φ : CG −→ End(Φ)
which for group elements g ∈ G is defined as
X
X
X
yx · hx ax ,
g · x · ax =
x · ax =
φ(g)
x∈X
x∈X
for g · x = yx · hx ,
x∈X
thus supplying Φ with the structure of a left CG-module as well. One refers
to Φ as the self-similarity bimodule, although in our terminology it is more
rightfully a C*-correspondence (lacking the left sided inner product in order
for it to be called a bimodule).
One would like to complete the algebra CG in such a way that the selfsimilarity bimodule Φ would become a Hilbert bimodule, thus allowing us
to speak of the C*-correspondence (Φ, φ). Such a completion of CG is called
a self-similar completion. Among such completions, there exists a certain
unique minimal completion, which is defined using the notion of generic
points.
Definition 6.1. A point w ∈ X ω is called G-generic if for every g ∈ G one
has either g(w) 6= w, or there exists a neighborhood U of w, such that every
point in U is fixed by g.
35
Denoting by G(w) the G-orbit of a fixed G-generic point w ∈ X ω , one
introduces the so called permutation representation
πw of CG on l2 (G(w)),
P
which is defined, for g ∈ G, f ∈ l2 (G(w)), f = u∈S αu u, αu ∈ C, S ⊂ G(w)
a finite subset,
X
πw (g)f =
αu g(u)
u∈S
and then extended to CG by linearity and continuity. Then we define a norm
|| · ||w on CG by ||a||w = ||πw (a)|| as the operator norm using the representation πw , for a ∈ CG. The completion of (CG, || · ||w ) is denoted AΦ . It is
shown in [2] that AΦ is a self-similar completion of CG (and not depending
on w), i.e. Φ becomes a Hilbert C*-module over AΦ , hence allowing us to
work with the C*-correspondence (Φ, φ) over AΦ .
In order to conclude that OΦ is simple, we can by Theorem 5.7 show
that Φ is minimal and nonperiodic.
To address minimality, we introduce the following notation for convenience,
Sx : AΦ −→ AΦ
Sx (a) = hx · e, a(x · e)iAΦ
Then invariance of an ideal J ⊆ AΦ , namely hΦ, JΦiAΦ ⊆ J, in particular
means Sx (J) ⊆ J for all x ∈ X. The operators Sx are bounded, since by
1
1
writing ξ = x · e, we have ||ξ|| = ||hξ, ξiAΦ || 2 = ||e∗ e|| 2 = ||e|| = 1, and then
||Sx (a)|| = ||hξ, aξiAΦ || ≤ ||ξ|| · ||aξ|| ≤ ||ξ||2 ||a|| = ||a||
thus ||Sx || ≤ 1.
Let w ∈ X ω be a G-generic point, meaning that for each g ∈ G, either
g(w) 6= w, or there exists a neighborhood of w which is fixed by g. In the
following presume w = x1 x2 x3 . . ..
Proposition 6.2. Let g ∈ G. Then Sxn · · · Sx1 (g) 6= 0 for all n ∈ N if and
only if g(w) = w.
Proof. Sx1 (g) 6= 0 is equivalent to g · x1 = x1 · h1 , for some h1 ∈ G, and then
Sx1 (g) = h1 . Sx2 (h1 ) 6= 0 is equivalent to h1 · x2 = x2 · h2 for some h2 ∈ G,
and so Sx2 (h1 ) = h2 . Continuing in this manner till the n-th iteration, we
get
g(x1 . . . xn . . .) = x1 h1 (x2 . . . xn . . .) = x1 x2 h2 (x3 . . . xn . . .) = . . . =
= x1 . . . xn hn . . .
It follows that Sxn · · · Sx1 (g) 6= 0 for all n if and only if g(w) = w.
36
It follows from Proposition 6.2 that for an element g ∈ G, if g(w) 6= w
then there exists m ∈ N such that Sxm · · · Sx1 (g) = 0. In the situtation that
g(w) = w, it would be beneficiary to know what happens when successive
applications of the maps Sx are done to g, other than it being non-zero.
Proposition 6.3. Let g ∈ G. If g(w) = w then there exists m ∈ N such
that Sxm · · · Sx1 (g) = e.
Proof. The point w ∈ X ω was G-generic, so the assumption g(w) = w means
there exists a neighborhood of w which is fixed by g. Cylindrical sets constitute a neighborhood basis, thus there exists a cylindrical set x1 . . . xm X ω
containing w and being kept fixed under g. Writing g · xi = xi · hi , for
i = 1, . . . m, we get
g(x1 . . . xm y1 y2 . . .) = x1 h1 (x2 . . . xm y1 y2 . . .) = . . . = x1 . . . xm hm (y1 y2 . . .)
= x1 x2 . . . xm y1 y2 . . .
for all y1 y2 . . . ∈ X ω . Thus hm (y1 y2 . . .) = y1 y2 . . . for all y1 y2 . . . ∈ X ω ,
which implies hm = e due to the faithfulness of the action. Hence we get
Sxm · · · Sx1 (g) = e as was to be shown.
Having obtained AΦ by representing CG on l2 (G(w)) and completing,
we now define a state φw : AΦ −→ AΦ
φw (a) = haδw , δw il2 .
It is clear that for b ∈ CG, φw picks out the
of isotropy
P coefficients in frontP
group elements. More precisely, for b = i bi gi , then φw (b) = i:gi ∈Gw bi .
The state φw may thus be advantegously used in conjunction with the preceding propositions. Also note that for an ideal J ⊆ AΦ one has φw |J = 0 if
and only if πw (a)δw = 0 for all a ∈ J, which is equivalent to πw (a)πw (b)δw =
0 for all a ∈ J, b ∈ AΦ , which in turn holds if and only if πw (a) = 0, i.e.
J itself is zero. Hence there do exist elements in J on which φw will be
nonzero.
Lemma 6.4. For every b ∈ CG there exists m ∈ N such that Sxm · · · Sx1 (b) =
φw (b)e.
P
Proof. Assume b = N
i=1 bi gi . Define the sets I1 = {i : gi (w) 6= w} and
I2 = {j : gj (w) = w}. From Proposition 6.2 it follows that for each i ∈ I1
there exists n(i) ∈ N such that Sxn(i) · · · Sx1 (gi ) = 0, and likewise from
Proposition 6.3 it follows that for each j ∈ I2 there exists m(j) ∈ N such
that Sxm(j) · · · Sx1 (gj ) = e. Put m = max{n(i), m(j) : i ∈ I1 , j ∈ I2 }. Then
for each gi we have
0 if i ∈ I1 ,
Sxm · · · Sx1 (gi ) =
e if i ∈ I2 .
37
Hence
N
X
X
bi )e = φw (b)e.
bi gi ) = (
Sxm · · · Sx1 (b) = Sxm · · · Sx1 (
i=1
i∈I2
Proposition 6.5. Φ is minimal.
Proof. Let {0} =
6 J ⊆ AΦ be an ideal such that hΦ, JΦiAΦ ⊆ J, hence also
Sx (J) ⊆ J for all x ∈ X. Let a∗ a ∈ J be such that φw (a∗ a) > P
0. For an
arbitrary ǫ > 0 find b ∈ CG such that ||a∗ a − b|| < ǫ. Suppose b = N
i=1 bi gi .
Then we also have |φw (a∗ a) − φw (b)| < ǫ. By Lemma 6.4 there exists m ∈ N
such that Sxm · · · Sx1 (b) = φw (b)e. The operators Sx are contractions, so we
have ||Sxm · · · Sx1 (a∗ a) − Sxm · · · Sx1 (b)|| < ǫ. It follows that
||Sxm · · · Sx1 (a∗ a) − φw (a∗ a)e|| ≤
||Sxm · · · Sx1 (a∗ a) − φw (b)e|| + ||φw (b)e − φw (a∗ a)e|| = 2ǫ
This shows that the sequence {Sxm · · · Sx1 (a∗ a)}m∈N ⊆ J converges to the
scalar multiple φw (a∗ a)e, hence φw (a∗ a)e ∈ J and thus J must be the whole
algebra AΦ .
We turn to discuss the nonperiodicity of the bimodule Φ. Let’s assume
that we have a unitary u : Φ −→ AΦ , and denote for shorthand ux = u(x·e),
for x ∈ X. These elements clearly satisfy
u∗x ux = e,
u∗x uy = 0 for x 6= y,
gux = uy h for g · x = y · h.
For a finite word v ∈ X ∗ define the map Tv : w 7→ vw, with partially defined
inverse Tv∗ : vw 7→ w.
Definition 6.6. A point w ∈ X ω is called strictly G-generic if for any v,
u ∈ X ∗ and any g ∈ G the transformation Tv gTu∗ either moves the point
w, or fixes w together with every point in a neighborhood of w, or is not
defined on w.
Let w = {xi }i ∈ X ω be a strictly G-generic point. We claim that for
any g ∈ G there exists n ∈ N such that the product
u∗xn · · · u∗x2 u∗x1 gux2 ux3 · · · uxn = 0.
Assume to the contrary that for all n the above product is nonzero. Clearly
this is the case if and only if
g · x2 = x1 · g2 ,
g2 · x3 = x2 · g3 ,
gi · xi+1 = xi · gi+1 ,
for some g2 ∈ G, with
for some g3 ∈ G, and so on
for all i ≥ 2, for elements gi ∈ G.
38
This means that g : x2 x3 x4 . . . 7−→ x1 x2 x3 x4 . . .. The point w was strictly
G-generic, and since the transformation gTx∗1 fixes w, we know there exists a neighborhood of w which is kept fixed by gTx∗1 , and in particular
some cylindrical set x1 . . . xm X ω which will be fixed. Taking any element
x1 . . . xm ym+1 ym+2 . . . from this cylindrical set, we compute
gTx∗1 (x1 . . . xm ym+1 ym+2 . . .) = g(x2 . . . xm ym+1 ym+2 . . .)
x1 x2 . . . xm gm+1 (ym+1 ym+2 . . .) = x1 x2 . . . xm ym+1 ym+2 . . .
which, because of the faithfulness of the group action, implies that gm+1 = e.
Hence xm = xm+i for all i ≥ 0 but this contradicts the fact that the point
w was strictly G-generic, ergo there must exist some n such that the above
product is zero.
Proposition 6.7. Φ is nonperiodic.
Proof. Assume that Φ is periodic. It suffices to consider periodicity for
n = 1, i.e. Φ ≈ AΦ (the general case amounts to replacing the basis X by
X n ). Then from the discussion above, extended from elements g ∈ G to
general elements of CG, we know that for any a ∈ CG there exists n such
that
u∗xn u∗xn−1 · · · u∗x2 aux1 ux2 · · · uxn = 0.
But on the other hand, for the element a = u∗x1 we have
u∗xn u∗xn−1 · · · u∗x2 u∗x1 ux1 ux2 · · · uxn = e,
which is absurd. Hence Φ cannot be periodic.
We are now able to give an alternative proof of Theorem 8.3 in [2].
Theorem 6.8. OΦ is simple.
Proof. We have established in Proposition 6.5 and Proposition 6.7 that Φ
is minimal and nonperiodic, hence it follows from Theorem 5.7 that OΦ is
simple.
39
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40