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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.
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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'
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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)
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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) · · ·...
......
.... . .
.
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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‖.
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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.
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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).
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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?
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms
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
B. K. Kwa±niewski Crossed products by arbitrary endomorphisms