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C ∗-correspondence functoriality of Cuntz-Pimsner algebras

Menevs ¸e Ery¨ uzl¨ u

September, 2020

What/Why are we doing?

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX Cross products by automorphisms

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX Cross products by automorphisms Cuntz algebras

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX Cross products by automorphisms Cuntz algebras Cuntz-Krieger algebras

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX Cross products by automorphisms Cuntz algebras Cuntz-Krieger algebras Graph algebras of graphs with sinks

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

What/Why are we doing?

AXA

OX Cross products by automorphisms Cuntz algebras Cuntz-Krieger algebras Graph algebras of graphs with sinks Crossed products by partial automorphisms

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Motivation

AXA and BYB are called Morita equivalent if there is an

imprimitivity bimodule AMB such that AX⊗AMB ∼ =AM⊗BYB.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Motivation

AXA and BYB are called Morita equivalent if there is an

imprimitivity bimodule AMB such that AX⊗AMB ∼ =AM⊗BYB. If the injective C∗-correspondences AXA and BYB are Morita equivalent then the corresponding Cuntz-Pimsner algebras OX and OY are Morita equivalent.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Tensor Product

Let AXB and BYC be C∗-correspondences. The algebraic tensor product X ⊙ Y has the A − C bimodule structure: a(x ⊗ y)c = ax ⊗ yc for a ∈ A, x ∈ X, y ∈ Y , c ∈ C, and the unique C-valued sesquilinear form x ⊗ y, u ⊗ vC =

y, x, uBvC

for x, u ∈ X, y, v ∈ Y . The Hausdorff completion is an A − C correspondence X ⊗B Y (with the left action a → ϕX(a)⊗A1Y ).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Definition 0.1 A representation of AXA on a C∗-algebra B is a pair consisting of a homomorphism π : A → B and a linear map t : X → B satisfying

1 t(x)∗t(y) = π(x, yA) , 2 π(a)t(x) = t(ϕ(a)x) , 3 t(x)π(a) = t(xa).

Also, there is a homomorphism ψt : K(X) → B with ψt(θx,y) = t(x)t(y)∗.

B A X t K(X) ψ π

Denote C∗(π, t) the C∗- algebra generated by the images of π and t in B.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

C ∗-algebra generated by representations

Consider AXA with the associated hom ϕ and a representa- tion (π, t). Define X ⊗0 = A, X ⊗1 = X, X ⊗n = X⊗AX ⊗n−1. Each X ⊗n is a C∗- correspondence over A with the left action defined by , ϕn(a)(x1⊗Ax2⊗A....xn) := ϕ(a)x1⊗A....⊗Axn. Now, set t0 = π and t1 = t. For n = 2, 3, ..., define a linear map tn : X ⊗n → C∗(π, t) by tn(x⊗Ay) = t(x)tn−1(y) for x ∈ X and y ∈ X ⊗n−1. Each (tn, π) is a representation of the C∗-correspondence X ⊗n and we have that C∗(π, t)=span{tn(x)tm(y)∗| x ∈ X ⊗n, y ∈ X ⊗m, n, m ∈ N}.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Fock Space!!!

F(X) =

∞

• n=0

X ⊗n = A ⊕ X ⊕ X ⊗2 ⊕ X ⊗3 ⊕ .... ={x = (x(n)) ∈

n X ⊗n : nx(n), x(n)A converges .}

For any y ∈ X, define t∞(y) ∈ L(F(X)) by t∞(y)(a, x1, x2, ....)=(0, ya, y⊗Ax1, y⊗Ax2, ....). Define the *-homomorphism ϕ∞ : A → L(F(X)) by ϕ∞(a) = diag(a, ϕ(a), ϕ2(a), ....)

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t∞(y) behaves : t∞(y)∗(a) = 0, t∞(y)∗(z⊗Axn) = ϕn(y, zA)(xn), for xn ∈ X ⊗n. So, t∞(y)∗(a, x1, x2⊗Ax′

2, x3⊗Ax′ 3⊗Ax′′ 3 , ....) =

(y, x1A, ϕ(y, x2(x′

2), ϕ2(y, x3(x′ 3⊗Ax′′ 3 ), ......).

Now, (t∞, ϕ∞) is a representation of AXA on L(F(X)), which is called the Fock representation. Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ∞ defined as above.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t∞(y) behaves : t∞(y)∗(a) = 0, t∞(y)∗(z⊗Axn) = ϕn(y, zA)(xn), for xn ∈ X ⊗n. So, t∞(y)∗(a, x1, x2⊗Ax′

2, x3⊗Ax′ 3⊗Ax′′ 3 , ....) =

(y, x1A, ϕ(y, x2(x′

2), ϕ2(y, x3(x′ 3⊗Ax′′ 3 ), ......).

Now, (t∞, ϕ∞) is a representation of AXA on L(F(X)), which is called the Fock representation. Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ∞ defined as above. C∗(ϕ∞, t∞)=span{tn

∞(xn)tm ∞(ym)∗ : xn ∈ X ⊗n, ym ∈ X ⊗m, n, m ∈ N},

which is isomorphic to TX !!!

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t∞(y) behaves : t∞(y)∗(a) = 0, t∞(y)∗(z⊗Axn) = ϕn(y, zA)(xn), for xn ∈ X ⊗n. So, t∞(y)∗(a, x1, x2⊗Ax′

2, x3⊗Ax′ 3⊗Ax′′ 3 , ....) =

(y, x1A, ϕ(y, x2(x′

2), ϕ2(y, x3(x′ 3⊗Ax′′ 3 ), ......).

Now, (t∞, ϕ∞) is a representation of AXA on L(F(X)), which is called the Fock representation. Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ∞ defined as above. C∗(ϕ∞, t∞)=span{tn

∞(xn)tm ∞(ym)∗ : xn ∈ X ⊗n, ym ∈ X ⊗m, n, m ∈ N},

which is isomorphic to TX !!! Side Note: By using ϕ∞ : A → TX, we get A(TX)TX .

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Here is how the adjoint of t∞(y) behaves : t∞(y)∗(a) = 0, t∞(y)∗(z⊗Axn) = ϕn(y, zA)(xn), for xn ∈ X ⊗n. So, t∞(y)∗(a, x1, x2⊗Ax′

2, x3⊗Ax′ 3⊗Ax′′ 3 , ....) =

(y, x1A, ϕ(y, x2(x′

2), ϕ2(y, x3(x′ 3⊗Ax′′ 3 ), ......).

Now, (t∞, ϕ∞) is a representation of AXA on L(F(X)), which is called the Fock representation. Notice that the Fock space can be viewed as a full A − A correspondence with the non-degenerate homomorphism ϕ∞ defined as above. C∗(ϕ∞, t∞)=span{tn

∞(xn)tm ∞(ym)∗ : xn ∈ X ⊗n, ym ∈ X ⊗m, n, m ∈ N},

which is isomorphic to TX !!! Side Note: By using ϕ∞ : A → TX, we get A(TX)TX .

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Katsura Ideal

Definition 0.2 For a C∗-correspondence X over A, define an ideal JX of A to be ϕ−1(K(X)) ∩ (Kerϕ)⊥ . Note that JX is the largest ideal on which the restriction of ϕ is an injection into K(X). F(X)JX is a Hilbert JX -module, and we have K(F(X)JX) = span{θζa,η ∈ K(F(X)) : ζ, η ∈ F(X), a ∈ JX}, which is an ideal of L(F(X)). In fact, it is an ideal of TX !!!

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

There comes the Cuntz-Pimsner algebra...

Let ρ : L(F(X)) → L(F(X)/K(F(X)JX) be the quotient map and set φ = ρ ◦ ϕ∞ and t = ρ ◦ t∞. Then,

1 (φ, t) is a covariant representation of AXA on

L(F(X)/K(F(X)JX))

2 This representation is injective. Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

There comes the Cuntz-Pimsner algebra...

Let ρ : L(F(X)) → L(F(X)/K(F(X)JX) be the quotient map and set φ = ρ ◦ ϕ∞ and t = ρ ◦ t∞. Then,

1 (φ, t) is a covariant representation of AXA on

L(F(X)/K(F(X)JX))

2 This representation is injective.

Katsura proves that C∗(φ, t) ∼ = OX. Hence, OX is the quotient C∗-algebra TX/K(F(X)JX).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

There comes the Cuntz-Pimsner algebra...

Let ρ : L(F(X)) → L(F(X)/K(F(X)JX) be the quotient map and set φ = ρ ◦ ϕ∞ and t = ρ ◦ t∞. Then,

1 (φ, t) is a covariant representation of AXA on

L(F(X)/K(F(X)JX))

2 This representation is injective.

Katsura proves that C∗(φ, t) ∼ = OX. Hence, OX is the quotient C∗-algebra TX/K(F(X)JX).

A(OX)OX

and

TX (OX)OX

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

CORES

Recall, ψtn : K(X ⊗n) → C∗(π, t) is defined by tn(xn)tn(yn)∗. Set B0 := π(A) and Bn = ψtn(K(X ⊗n)) ⊂ C∗(π, t). For m, n ∈ N with m ≤ n, define B[m, n] ⊂ C∗(π, t) by B[m, n] = Bm + Bm+1 + .... + Bn. For m ∈ N, define C∗-subalgebra B[m, ∞) of C∗(π, t) by B[m, ∞) =

∞

• n=m

B[m, n]. B[0, ∞) is called the CORE of C∗(π, t).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

1 Objects: C*-algebras Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

1 Objects: C*-algebras 2 Morphisms: Isomorphism classes of C*-correspondences

For an A − B correspondence X, we mean right Hilbert B-module X with a non-degenerate homomorphism ϕX : A → L(X).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

1 Objects: C*-algebras 2 Morphisms: Isomorphism classes of C*-correspondences

For an A − B correspondence X, we mean right Hilbert B-module X with a non-degenerate homomorphism ϕX : A → L(X).

3 Composition: Isomorphism class of the balanced tensor

product. Consider [AXB] and [BYC]. Then the composition is [AX⊗BYC]

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

1 Objects: C*-algebras 2 Morphisms: Isomorphism classes of C*-correspondences

For an A − B correspondence X, we mean right Hilbert B-module X with a non-degenerate homomorphism ϕX : A → L(X).

3 Composition: Isomorphism class of the balanced tensor

product. Consider [AXB] and [BYC]. Then the composition is [AX⊗BYC]

4 The identity morphism on A is [AAA] Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Enchilada Category

1 Objects: C*-algebras 2 Morphisms: Isomorphism classes of C*-correspondences

For an A − B correspondence X, we mean right Hilbert B-module X with a non-degenerate homomorphism ϕX : A → L(X).

3 Composition: Isomorphism class of the balanced tensor

product. Consider [AXB] and [BYC]. Then the composition is [AX⊗BYC]

4 The identity morphism on A is [AAA] 5 Zero morphism : [A0B] Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Enchilada Category of C ∗-correspondences

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Enchilada Category of C ∗-correspondences

Objects are non-degenerate C∗-correspondences

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Enchilada Category of C ∗-correspondences

Objects are non-degenerate C∗-correspondences Morphisms from AXA to BYB are the isomorphism classes of regular A − B correspondences [AMB] such that X⊗AM ∼ = M⊗BY as A − B correspondences. A A B B

X M M Y

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Enchilada Category of C ∗-correspondences

Objects are non-degenerate C∗-correspondences Morphisms from AXA to BYB are the isomorphism classes of regular A − B correspondences [AMB] such that X⊗AM ∼ = M⊗BY as A − B correspondences. A A B B

X M M Y

F(X)⊗AM ∼ = M⊗BF(Y ) Composition is the same as in the enchilada category.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Enchilada Category of C ∗-correspondences

Objects are non-degenerate C∗-correspondences Morphisms from AXA to BYB are the isomorphism classes of regular A − B correspondences [AMB] such that X⊗AM ∼ = M⊗BY as A − B correspondences. A A B B

X M M Y

F(X)⊗AM ∼ = M⊗BF(Y ) Composition is the same as in the enchilada category. Identity morphism on AXA is [AAA].

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Functor

ECCor

AXA AXA [AMB]

− − − → BYB Enchilada Category OX [OX (F(X)⊗AM⊗BOY )OY ]

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Outline

Theorem 0.3

• F(X)⊗AM⊗BB[0, ∞)/

F(X)⊗AM⊗BB[1, ∞)) is an OX − B

• correspondence. (Noting that B[0, ∞) represents the core of BYB).

Proposition 0.4

TX

F(X)⊗AM⊗BOY

• OY

∼ = TX

Z/

ZB[1, ∞)⊗BOY

• OY

Corollary 0.5 Let AXA and BYB be objects in the enchilada category of C∗-correspondences and [AMB] be a morphism between them. Then, F(X)⊗AM⊗BOY is an OX-OY correspondence.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

C ∗-correspondence isomorphism

AXB ∼

= AYB if there is a linear bijection U : X → Y such that U(ax) = aU(x) Ux, UyB = x, yB for all a ∈ A, and x, y ∈ X.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

An A − B bimodule X0 is called a pre-correspondence if it has a B-valued semi-inner product satisfying x, y · bB = x, yBb x, y∗

B = y, xB

a · x, a · xB ≤ a2 x, xB for all a ∈ A, b ∈ B and, x, y ∈ X0.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

An A − B bimodule X0 is called a pre-correspondence if it has a B-valued semi-inner product satisfying x, y · bB = x, yBb x, y∗

B = y, xB

a · x, a · xB ≤ a2 x, xB for all a ∈ A, b ∈ B and, x, y ∈ X0. Hausdorff completion X of X0 becomes an A − B correspondence.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

An A − B bimodule X0 is called a pre-correspondence if it has a B-valued semi-inner product satisfying x, y · bB = x, yBb x, y∗

B = y, xB

a · x, a · xB ≤ a2 x, xB for all a ∈ A, b ∈ B and, x, y ∈ X0. Hausdorff completion X of X0 becomes an A − B correspondence. For AXB and BYC, X ⊙ Y is a A − C pre-correspondence !

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Let Z be an A − B correspondence and assume there is a map Φ : X0 → Z satisfying Φ(a · x) = ϕZ(a)Φ(x) and Φ(x), Φ(y)B = x, yB. Then, Φ extends uniquely to an injective A − B correspondence homomorphism Φ : X → Z.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Lemma 0.6 Let A, B be C∗-algebras with A ⊂ B and let X be a B − C

• correspondence. Then, AAB⊗BXC ∼

= AAX C. Proof.

1 Let Φ : AB ⊙ X → AX be the unique linear map defined by

n

i=1 aibi⊗Bxi → n i=1 ϕ(aibi)xi.

2 Φ preserves the left action 3 Φ preserves the inner product 4 Φ is surjective 5 Φ extends to the isomorphism AB⊗BX → AX. Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Some important factorizations

A(F(X)⊗AF(X))A ∼

= AF(X)A

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Some important factorizations

A(F(X)⊗AF(X))A ∼

= AF(X)A

A(OX)OX ∼

= A(TX⊗TX OX)OX

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Some important factorizations

A(F(X)⊗AF(X))A ∼

= AF(X)A

A(OX)OX ∼

= A(TX⊗TX OX)OX

OX (F(X)⊗AOX)OX ∼

= OX (OX)OX

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Left action of a quotient C ∗-algebra

Let X be an A − B correspondence and I is an ideal of A such that I ⊂ Ker(ϕX). Then, we may view X as an A/I − B correspondence:

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Left action of a quotient C ∗-algebra

Let X be an A − B correspondence and I is an ideal of A such that I ⊂ Ker(ϕX). Then, we may view X as an A/I − B correspondence: Let q : A → A/I be the quotient map. Define a linear map Φ : A/I → L(X) so that Φ(q(a)) = ϕX(a) for any a ∈ A.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Let AXB, and the ideal J of B be given. q : B → B/J and π : X → X/XJ Then, X/XJ is a Hilbert B/J-module with the operations π(x)q(b) := π(xb) π(x), π(y)B/J := q(x, yB).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Let AXB, and the ideal J of B be given. q : B → B/J and π : X → X/XJ Then, X/XJ is a Hilbert B/J-module with the operations π(x)q(b) := π(xb) π(x), π(y)B/J := q(x, yB). Define a linear map β : LB(X) → LB/J(X/XJ) by β(T)(π(x)) := π(T(x)). X/XJ becomes an A − B/J correspondence with β ◦ ϕ : A → LB(X) → LB/J(X/XJ).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Quotient C ∗-correspondence

A(X⊗BB/I)B/I ∼

= AX/XIB/I.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Quotient C ∗-correspondence

A(X⊗BB/I)B/I ∼

= AX/XIB/I. Proposition 0.7 Let I be an ideal of A such that I ⊂ Ker(β ◦ ϕ) we may view X/XJ as an A/I − B/J correspondence.

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

Going back to the main theorem

Theorem 1.1

• F(X)⊗AM⊗BB[0, ∞)/

F(X)⊗AM⊗BB[1, ∞) is an OX − B

correspondence. strategy for the proof Denote Z := F(X)⊗AM⊗BB[0, ∞). Consider the homomorphism ξ : TX → L(Z/ZB[1, ∞)). Prove K(F(X)JX) ⊂ Ker(ξ).

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Functor

AXA AXA [AMB]

− − − → BYB OX OX

[F(X)⊗AM⊗BOY ]

− − − − − − − − − − − → OY

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

The Functor

AXA AXA [AMB]

− − − → BYB OX OX

[F(X)⊗AM⊗BOY ]

− − − − − − − − − − − → OY It preserves the composition. In ECCor, identity morphism on given AXA is [AAA] and it is mapped to [OX (F(X)⊗AA⊗AOX)OX ] which equals to [OX (OX)OX ].

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras

THANK YOU

Menevs ¸e Ery¨ uzl¨ u C∗-correspondence functoriality of Cuntz-Pimsner algebras