crossed products by arbitrary endomorphisms

Banach Algebras and Applications, Gothenburg 29.07.2013

Crossed products by arbitrary endomorphisms

Bartosz Kosma Kwa±niewskiUniversity of Biaªystok (soon IMPAN Warsaw)

Check out also my poster

on Wednesday !!!

B. K. Kwa±niewski, A. V. Lebedev �Crossed products by endomorphismsand reduction of relations in relative Cuntz-Pimsner algebras�J. of Funct. Analysis, 264 (2013), no. 8, 1806-1847

B. K. Kwa±niewski Crossed products by arbitrary endomorphisms

Question of the Day: What is crossed product?

Throughout A is unital C ∗-algebra.

Crossed product by an automorphism α : A→ A is a universalC ∗-algebra C ∗(A, u) generated by A and u subject to relations:

α(a) = uau∗, α−1(a) = u∗au, a ∈ A

Problem If α : A→ A is an endomorphism, then α−1(a) = u∗au.What relation should we use instead?

Let A ⊂ B be C ∗-algebras with a common unit 1, U ∈ B .

Proposition (the Hint).

Let α : A→ A be a map of the form α(a) = UaU∗. Then

α is an endomorphism ⇐⇒ U partial isometry, U∗U ∈ A′,

where A′ is the commutant of A.


Towards the crossed product construction

Assume α(a) = UaU∗ is an endomorphism of A.

Proposition.

1)C ∗(A,U) = span{U∗naUm : a ∈ A, n,m ∈ N}

2)J = {a ∈ A : U∗Ua = a} = U∗UA ∩ A

is an ideal in A such that J ∩ kerα = {0} (J ⊂ (kerα)⊥)

Rem. In the crossed product construction

the elements U∗naUm are the 'bricks'

the ideal J = {a ∈ A : U∗Ua = a} is the 'cement'


1) Reading o� the algebraic structure from C∗(A,U)

Consider in�nite matrices with entries labeled by N = {0, 1, 2, ...}M(A) := {[an,m] : an,m ∈ A only �nite entries non zero}

and a mapping Ψ :M(A)→ C∗(A,U) given by

Ψ([an,m]) =∑

n,m∈NU∗nan,mU

m

Proposition. The map Ψ becomes a ∗-homomorphism if

we de�ne onM(A) the ∗-algebra structure (M(A),+, ·, ∗, ?) as follows

(a + b)m,n := am,n + bm,n, (1)

(λa)m,n := λam,n (2)

(a∗)m,n := a∗n,m (3)

and

a ? b := a ·∞∑j=0

Λj(b) +∞∑j=1

Λj(a) · b (4)


M(A) := {[an,m] : an,m ∈ A only �nite entries non zero}

Proposition. The map Ψ becomes a ∗-homomorphism if

we de�ne onM(A) the ∗-algebra structure (M(A),+, ·, ∗, ?) as follows

(a + b)m,n := am,n + bm,n (1)

(λa)m,n := λam,n (2)

(a∗)m,n := a∗n,m (3)

and

a ? b = a ·∞∑j=0

Λj(b) +∞∑j=1

Λj(a) · b (4)

where · is matrix multiplication and Λ :M(A)→M(A) is given by

Λ([an,m]) :=

0 0 0 0 · · ·0 α(a0,0) α(a0,1) α(a0,2) · · ·0 α(a1,0) α(a1,1) α(a1,2) · · ·0 α(a2,0) α(a2,1) α(a2,2) · · ·...

......

.... . .

.


2) Calculation of norm of elements in C ∗(A,U)

An element [an,m] ∈M(A) is k-diagonal, where k ∈ Z, if it is of the form

if k ≥ 0, or

0

0

ak,0

ar+k,r

{kr + 1

0

0

a0,−k

ar+k,r

−k︷︸︸︷ r+k+1︷︸︸︷, if k < 0.

Proposition. If a = Ψ([an,m]) where [an,m] is k-diagonal, then

‖a‖ = limn→∞

max

maxi=1,...,n

{d( i∑

j=0,j+k≥0

αi−j (aj+k,j ), J)}, d(an+k,n, kerα)

(5)

where J = {a ∈ A : U∗Ua = a} and d(a, I ) = infb∈I ‖a− b‖.


Theorem (crossed product construction)

Let α : A→ A be an endomorphism and J ⊂ (kerα)⊥. There is a uniqueC∗-seminorm ‖ · ‖ on the ∗-algebra (M(A),+, ·, ∗, ?) such that (5) holdsand either

1) ‖∑k

ak‖ = ‖∑k

λkak‖, for all λ ∈ T,

or

2) ‖a0‖ ≤ ‖∑k

ak‖

for all k-diagonal elements ak ∈M(A). This C∗-seminorm yields aC∗-algebra

C∗(A, α, J) :=M(A)/‖ · ‖,which is generated by the elements

u :=

0 α(1) 0 · · ·0 0 0 · · ·0 0 0 · · ·...

.

.

.

.

.

....

and a :=

a 0 0 · · ·0 0 0 · · ·0 0 0 · · ·...

.

.

.

.

.

....

, a ∈ A,

and universal subject to relations: α(a) = uau∗, J = {a ∈ A : u∗ua = a}.B. K. Kwa±niewski Crossed products by arbitrary endomorphisms

De�nition Let α : A→ A be an endomorphism and J ⊂ (kerα)⊥.

We call C∗(A, α, J) the crossed product of A by α relative to J.If J = (kerα)⊥ we write C∗(A, α) and call it crossed product of A by α.

C∗(A, α, J) is a universal C∗-algebra generated by A and u subject to

α(a) = uau∗, a ∈ A, J = {a ∈ A : u∗ua = a} = u∗uA ∩ A.

Proposition

kerα unital =⇒ C∗(A, α) is universal generated by A and u subject to

α(a) = uau∗, a ∈ A, u∗u ∈ A.

α monomorphism =⇒ C∗(A, α) = Aoα N Stacey's crossed product:

α(a) = uau∗, a ∈ A, u∗u = 1.

α automorphism =⇒ C∗(A, α) = Aoα Z classical crossed product:

α(a) = uau∗, α−1(a) = u∗au, a ∈ A.


Relevant constructions

J-Reduction of an endomorphism

Suppose J is an arbitrary ideal in A.Let J∞ be the smallest α-invariantideal s.t. putting q : A→ A/J∞ and

Ar := q(A), αr ◦q := q◦α, Jr := q(J).

we have Jr ⊂ (kerαr )⊥.

C∗(Ar , αr , Jr ) is generated by an imageof A and u subject to relations

α(a) = uau∗, a ∈ A, J ⊂ u∗uA ∩ A

J-Unitization of kernel

If J ⊂ (kerα)⊥ one can construct anendomorphism αJ : AJ → AJ such that

1) A ⊂ AJ , αJ |A = α, kerαJ is unital

2) AJ = A ⇐⇒(

kerα is unitalJ = (kerα)⊥

)3) C∗(A, α, J) ∼= C∗(AJ , αJ)

αJ(a) = uau∗, a ∈ AJ , u∗u ∈ AJ

Hereditation of range. Suppose kerα is unital.

There is an endomorphism β : B → B extending α : A→ A such thatker β is unital, β(B) is hereditary and C∗(A, α) ∼= C∗(B, β).


Remarks on Exel's crossed product and topological freeness

Standing assumptions: kerα is unital and α(A) is hereditary in A.

Proposition [Kwa1, KL, ABL]

There is a unique non-degenerate transfer operator L : A→ A forα : A→ A and C ∗(A, α) is a universal C ∗-algebra generated by A

and u subject to:

α(a) = uau∗, L(a) = u∗au, a ∈ A.

Moreover, C ∗(A, α) ∼= Aoα,L N � Exel's crossed product.

Remark. α : L(A)→ α(A) is an isomorphism and its dual

α : α(A)→ L(A) may be treated as a partial homeomorphism of A:

A

α(A) L(A)α


A

α(A) L(A)α

Uniqueness Theorem [Kwa2]. If α is topologically free

(the set of periodic points of period n ∈ N has empty interior), then forany faithful representation π : A→ B and U ∈ B such that

π(α(a)) = Uπ(a)U∗, π(L(a)) = U∗π(a)U, a ∈ A,

the mappingsa 7→ π(a), a ∈ A, u 7→ U

yield the isomorphism C ∗(A, α) ∼= C ∗(π(A),U).

Open problem:

How to de�ne and topological freeness for an arbitraryendomorphism?


References

[KL] B. K. Kwa±niewski, A. V. Lebedev �Crossed products byendomorphisms and reduction of relations in relative Cuntz-Pimsneralgebras� J. of Functional Analysis (2013)

[Exel] R. Exel: "A new look at the crossed-product of a C∗-algebra by anendomorphism", Ergodic Theory Dynam. Systems, (2003)

[ABL] A.B. Antonevich, V.I. Bakhtin, A.V. Lebedev, "Crossed product ofC∗-algebra by an endomorphism, coe�cient algebras and transferoperators", Sb. Math. (2011)

[Kwa1] B. K. Kwa±niewski, �On transfer operators for C*-dynamicalsystems� Rocky J. Math. (2012),

[Kwa2] B. K. Kwa±niewski �Dynamical system dual to interactions andgraph algebras� arXiv:1301.5125

[Kwa3] B. K. Kwa±niewski � Extensions of C∗-dynamical systems tosystems with complete transfer operators" arXiv:OA/0703800


crossed products by arbitrary endomorphisms

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