KK -equivalence of amalgamated free products of C -algebras . Kei - - PowerPoint PPT Presentation

. .

KK-equivalence of amalgamated free products

• f C∗-algebras

Kei Hasegawa

Kyushu Univ.

July 27, 2016 Young Mathematicians in C∗-Algebras, University of M¨ unster

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Theorem (Cuntz (1982))

. . K0(C∗(Fn)) = Z, K1(C∗(Fn)) = Zn. . . . . . .

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.

Theorem (Cuntz (1982))

. . K0(C∗(Fn)) = Z, K1(C∗(Fn)) = Zn. .

Theorem (Pimsner–Voiculescu (1982))

. . ∀n ∈ N, ∀α = ∗n

k=1αk : Fn ↷ B, we have the exact sequence with

σ = ∑n

k=1(1 − αk∗)

⊕n

k=1 K0(B) σ

K0(B) K0(B ⋊α,r Fn)

• K1(B ⋊α,r Fn)
• K1(B)
• ⊕n

k=1 K1(B), σ

• .

Theorem (Pimsner–Voiculescu (1982))

. . K0(C∗

r (Fn)) = Z,

K1(C∗

r (Fn)) = Zn.

Note: C∗(Fn) and C∗

r (Fn) are non-isomorphic when n ≥ 2.

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We say that C∗-algebras B and C are KK-equivalent if ∃x ∈ KK(B, C) and ∃y ∈ KK(C, B) such that [idB] = y ◦ x and [idC] = x ◦ y. KK-equivalence implies K0(B) ∼ = K0(C) and K1(B) ∼ = K1(C). .

Definition (Cuntz (1983))

. . A discrete group Γ is K-amenable if one of the following equivalent conditions holds: (i) λ: C∗(Γ) → C∗

r (Γ) implements a KK-equivalence.

(ii) ∃t ∈ K 0(C∗

r (Γ)) = KK(C∗ r (Γ), C) such that [1Γ] = t ◦ [λ] in

K 0(C∗(Γ)) = KK(C∗(Γ), C). C∗(Γ)

1Γ

• λ
• C

C∗

r (Γ) t

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.

Definition (Cuntz (1983))

. . A discrete group Γ is K-amenable if one of the following equivalent conditions holds: (i) λ: C∗(Γ) → C∗

r (Γ) implements a KK-equivalence.

(ii) ∃t ∈ K 0(C∗

r (Γ)) = KK(C∗ r (Γ), C) such that [1Γ] = t ◦ [λ] in

K 0(C∗(Γ)) = KK(C∗(Γ), C). C∗(Γ)

1Γ

• λ
• C

C∗

r (Γ) t

• .

Theorem (Cuntz (1983), Julg–Valette (1984), Pimsner (1986))

. . Γ1 ∗Λ Γ2 is K-amenable if and only if so are both Γ1 and Γ2.

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K-homology group K 0(C∗(Γ))

A Fredholm Γ-module is a triplet (π0, π1, F) such that π0 : Γ ↷ H0 and π1 : Γ ↷ H1 are unitary representations. F : H0 → H1 is an operator s.t. 1−F ∗F and 1−FF ∗ are compact. Fπ0(g) − π1(g)F is compact for any g ∈ Γ, K 0(C∗(Γ)) is the additive group of homotopy equivalence classes of Fredholm Γ-modules: addition: direct sum, inverse: −[(π1, π2, F)] = [(π2, π1, F ∗)], zero element: [(π, Ad U ◦ π, U)] with U unitary. e.g. [1Γ] = [(1Γ, 0, 0)] K 0(C∗

r (Γ)) is defined by a similar way restricted to the representations

which are weakly contained in λ: Γ ↷ ℓ2(Γ).

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Julg–Valette’s geometric construction

Let Γ := Γ1 ∗Λ Γ2, and (V , E) = (Γ/Γ1 ⊔ Γ/Γ2, Γ/Λ) be the Bass–Serre tree for Γ.

gΓ1

• gΛ

gΓ2

• We have the unitary repns

π0 : Γ ↷ H0 := ℓ2(V ), π1 : Γ ↷ H1 = ℓ2(E).

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Julg–Valette’s geometric construction

Let Γ := Γ1 ∗Λ Γ2, and (V , E) = (Γ/Γ1 ⊔ Γ/Γ2, Γ/Λ) be the Bass–Serre tree for Γ.

gΓ1

• gΛ

gΓ2

• We have the unitary repns

π0 : Γ ↷ H0 := ℓ2(V ), π1 : Γ ↷ H1 = ℓ2(E). Define the bijection β : V \ {eΓ1} → E by β(eΓ2) = eΛ and β(g1g2 · · · gnΓk) = g1g2 · · · gn−1Λ if gn ∈ Γk+1(mod2).

x

• β(x)
• eΓ2
• eΓ1

eΛ

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Julg–Valette’s geometric construction

Define the co-isometry S : H0 → H1 by Sδx = δβ(x) and SδeΓ1 = 0. Then, (π0, π1, S) enjoys the axiom of Fredholm Γ-module: (i) 1 − S∗S is rank one and 1 − SS∗ = 0. (ii) Sπ0(g) − π1(g)S is compact for any g ∈ Γ, and so defines an element α := [(π0, π1, S)] ∈ K 0(C∗(Γ)). .

Theorem (Julg–Valette)

. .

α = [1Γ].

Remark: if Γ1, Γ2 are amenable, then π0 = λΓ/Γ1 ⊕ λΓ/Γ2 ≺ λ, π1 = λΓ/Λ ≺ λ, so α = t ◦ [λ] for some t ∈ K 0(C∗

r (Γ)). Thus, Γ is

K-amenable.

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Full amalgamated free product

A1 ⊃ D ⊂ A2: unital C∗-algebras. A := A1 ∗

D A2: the full (or universal) amalgamated free product.

D

• A1
• A2

A

Example: C∗(Γ1 ∗

Λ Γ2) = C∗(Γ1)

∗

C∗(Λ) C∗(Γ2)

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Reduced amalgamated free product

Assume ∃Ek : Ak → D “GNS-faithful” cond. expectation. The reduced amalgamated free product (A, E) := (A1, E1) ∗

D (A2, E2) is given by:

(i) E : A → D is a GNS-faithful cond. expectation. (ii) (Freeness): ∀n ∈ N, ∀i1 ̸= i2 ̸= · · · ̸= in ∈ {1, 2}, and ∀ak ∈ ker Eik, 1 ≤ k ≤ n, one has E(a1a2 · · · an) = 0. Example: (C∗

r (Γ1 ∗ Λ Γ2), E) = (C∗ r (Γ1), E1)

∗

C∗

r (Λ) (C∗

r (Γ2), E2)

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.

Main Theorem (H, Fima–Germain, ’15)

. . If A1 and A2 are separable, then the canonical surjection λ: A → A implements a KK-equivalence.

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.

Main Theorem (H, Fima–Germain, ’15)

. . If A1 and A2 are separable, then the canonical surjection λ: A → A implements a KK-equivalence. For (A, E) = (A1, E1) ∗D (A2, E2), we have EAk : A → Ak. Let (X, ϕX, ξ0), (Yk, ϕk, ηk) be the GNS-repns for E and Ek, resp. Consider the A-A C∗-correspondences (H0, π0) := ⊕

k=1,2

(Yk ⊗Ak A, ϕk ⊗ 1) “Γ/Γ1 ⊔ Γ/Γ2” (H1, π1) := (X ⊗D A, ϕX ⊗ 1) “Γ/Λ” We then find S : H0 → H1 such that ( H0 ⊕ H1, π0 ⊕ π1, [

S∗ S

]) implements α ∈ KK(A, A). A similar argument shows that [idA] = α ◦ [λ].

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.

Main Theorem (H, Fima–Germain, ’15)

. . If A1 and A2 are separable, then the canonical surjection λ: A → A implements a KK-equivalence. For (A, E) = (A1, E1) ∗D (A2, E2), we have EAk : A → Ak. Let (X, ϕX, ξ0), (Yk, ϕk, ηk) be the GNS-repns for E and Ek, resp. Consider the A-A C∗-correspondences (H0, π0) := ⊕

k=1,2

(Yk ⊗Ak A, ϕk ⊗ 1) “Γ/Γ1 ⊔ Γ/Γ2” (H1, π1) := (X ⊗D A, ϕX ⊗ 1) “Γ/Λ” We then find S : H0 → H1 such that ( H0 ⊕ H1, π0 ⊕ π1, [

S∗ S

]) implements α ∈ KK(A, A). A similar argument shows that [idA] = α ◦ [λ]. Also, one can show that [idA] = [λ] ◦ α.

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Theorem (Thomsen (2003))

. . Let ik : D ֒ → Ak and jk : Ak ֒ → A be inclusion maps. Then, we have the exact sequence sequence for any separable B: K0(D)

(i1∗,i2∗)

− − − − → K0(A1) ⊕ K0(A2)

j1∗−j2∗

− − − − → K0(A)

• 

  

• K1(A)

j1∗−j2∗

← − − − − K1(A1) ⊕ K1(A2)

(i1∗,i2∗)

← − − − − K1(D)

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