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

. KK -equivalence of amalgamated free products of C ∗ -algebras . Kei Hasegawa Kyushu Univ. July 27, 2016 Young Mathematicians in C ∗ -Algebras, University of M¨ unster 1 / 11

. . . . . . . Theorem (Cuntz (1982)) . K 1 (C ∗ ( F n )) = Z n . K 0 (C ∗ ( F n )) = Z , . 2 / 11

� � � � . Theorem (Cuntz (1982)) . K 1 (C ∗ ( F n )) = Z n . K 0 (C ∗ ( F n )) = Z , . . Theorem (Pimsner–Voiculescu (1982)) . ∀ n ∈ N , ∀ α = ∗ n k =1 α k : F n ↷ B, we have the exact sequence with σ = ∑ n k =1 (1 − α k ∗ ) σ ⊕ n � K 0 ( B ) � K 0 ( B ⋊ α, r F n ) k =1 K 0 ( B ) ⊕ n K 1 ( B ⋊ α, r F n ) K 1 ( B ) k =1 K 1 ( B ) , σ . . Theorem (Pimsner–Voiculescu (1982)) . r ( F n )) = Z n . K 0 (C ∗ r ( F n )) = Z , K 1 (C ∗ . Note: C ∗ ( F n ) and C ∗ r ( F n ) are non-isomorphic when n ≥ 2. 2 / 11

� � � We say that C ∗ -algebras B and C are KK -equivalent if ∃ x ∈ KK ( B , C ) and ∃ y ∈ KK ( C , B ) such that [id B ] = y ◦ x and [id C ] = x ◦ y . KK -equivalence implies K 0 ( B ) ∼ = K 0 ( C ) and K 1 ( B ) ∼ = K 1 ( 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 [1 Γ ] = t ◦ [ λ ] in K 0 (C ∗ (Γ)) = KK (C ∗ (Γ) , C ). 1 Γ C ∗ (Γ) C λ t C ∗ r (Γ) . 3 / 11

� � � . 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 [1 Γ ] = t ◦ [ λ ] in K 0 (C ∗ (Γ)) = KK (C ∗ (Γ) , C ). 1 Γ C ∗ (Γ) C λ t C ∗ r (Γ) . . Theorem (Cuntz (1983), Julg–Valette (1984), Pimsner (1986)) . Γ 1 ∗ Λ Γ 2 is K-amenable if and only if so are both Γ 1 and Γ 2 . . 4 / 11

K -homology group K 0 (C ∗ (Γ)) A Fredholm Γ -module is a triplet ( π 0 , π 1 , F ) such that π 0 : Γ ↷ H 0 and π 1 : Γ ↷ H 1 are unitary representations. F : H 0 → H 1 is an operator s.t. 1 − F ∗ F and 1 − FF ∗ are compact. F π 0 ( g ) − π 1 ( 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: − [( π 1 , π 2 , F )] = [( π 2 , π 1 , F ∗ )], zero element: [( π, Ad U ◦ π, U )] with U unitary. e.g. [1 Γ ] = [(1 Γ , 0 , 0)] K 0 (C ∗ r (Γ)) is defined by a similar way restricted to the representations which are weakly contained in λ : Γ ↷ ℓ 2 (Γ). 5 / 11

Julg–Valette’s geometric construction Let Γ := Γ 1 ∗ Λ Γ 2 , and ( V , E ) = (Γ / Γ 1 ⊔ Γ / Γ 2 , Γ / Λ) be the Bass–Serre tree for Γ. g Γ 1 g Λ g Γ2 • • We have the unitary repns π 0 : Γ ↷ H 0 := ℓ 2 ( V ) , π 1 : Γ ↷ H 1 = ℓ 2 ( E ) . 6 / 11

Julg–Valette’s geometric construction Let Γ := Γ 1 ∗ Λ Γ 2 , and ( V , E ) = (Γ / Γ 1 ⊔ Γ / Γ 2 , Γ / Λ) be the Bass–Serre tree for Γ. g Γ 1 g Λ g Γ2 • • We have the unitary repns π 0 : Γ ↷ H 0 := ℓ 2 ( V ) , π 1 : Γ ↷ H 1 = ℓ 2 ( E ) . Define the bijection β : V \ { e Γ 1 } → E by β ( e Γ 2 ) = e Λ and β ( g 1 g 2 · · · g n Γ k ) = g 1 g 2 · · · g n − 1 Λ if g n ∈ Γ k +1(mod2) . x e Γ 2 • • • • β ( x ) e Λ • • • e Γ 1 6 / 11

Julg–Valette’s geometric construction Define the co-isometry S : H 0 → H 1 by S δ x = δ β ( x ) and S δ e Γ 1 = 0. Then, ( π 0 , π 1 , S ) enjoys the axiom of Fredholm Γ-module: (i) 1 − S ∗ S is rank one and 1 − SS ∗ = 0. (ii) S π 0 ( g ) − π 1 ( g ) S is compact for any g ∈ Γ, and so defines an element α := [( π 0 , π 1 , S )] ∈ K 0 (C ∗ (Γ)). . Theorem (Julg–Valette) . α = [1 Γ ] . . Remark : if Γ 1 , Γ 2 are amenable, then π 0 = λ Γ / Γ 1 ⊕ λ Γ / Γ 2 ≺ λ , π 1 = λ Γ / Λ ≺ λ , so α = t ◦ [ λ ] for some t ∈ K 0 (C ∗ r (Γ)). Thus, Γ is K -amenable. 7 / 11

� � � Full amalgamated free product A 1 ⊃ D ⊂ A 2 : unital C ∗ -algebras. A := A 1 ∗ D A 2 : the full (or universal) amalgamated free product. D � � A 1 � � � � A 2 � � � A Example : C ∗ (Γ 1 ∗ Λ Γ 2 ) = C ∗ (Γ 1 ) C ∗ (Λ) C ∗ (Γ 2 ) ∗ 8 / 11

Reduced amalgamated free product Assume ∃ E k : A k → D “GNS-faithful” cond. expectation. The reduced amalgamated free product ( A , E ) := ( A 1 , E 1 ) ∗ D ( A 2 , E 2 ) is given by: (i) E : A → D is a GNS-faithful cond. expectation. (ii) (Freeness): ∀ n ∈ N , ∀ i 1 ̸ = i 2 ̸ = · · · ̸ = i n ∈ { 1 , 2 } , and ∀ a k ∈ ker E i k , 1 ≤ k ≤ n , one has E ( a 1 a 2 · · · a n ) = 0. Example : (C ∗ r (Γ 1 ∗ Λ Γ 2 ) , E ) = (C ∗ r (Γ 1 ) , E 1 ) r (Λ) (C ∗ ∗ r (Γ 2 ) , E 2 ) C ∗ 9 / 11

. Main Theorem (H, Fima–Germain, ’15) . If A 1 and A 2 are separable, then the canonical surjection λ : A → A implements a KK-equivalence. . 10 / 11

. Main Theorem (H, Fima–Germain, ’15) . If A 1 and A 2 are separable, then the canonical surjection λ : A → A implements a KK-equivalence. . For ( A , E ) = ( A 1 , E 1 ) ∗ D ( A 2 , E 2 ), we have E A k : A → A k . Let ( X , ϕ X , ξ 0 ), ( Y k , ϕ k , η k ) be the GNS-repns for E and E k , resp. Consider the A - A C ∗ -correspondences ⊕ ( H 0 , π 0 ) := ( Y k ⊗ A k A , ϕ k ⊗ 1) “Γ / Γ 1 ⊔ Γ / Γ 2 ” k =1 , 2 ( H 1 , π 1 ) := ( X ⊗ D A , ϕ X ⊗ 1) “Γ / Λ” We then find S : H 0 → H 1 such that ( [ S ∗ ]) H 0 ⊕ H 1 , π 0 ⊕ π 1 , S implements α ∈ KK ( A , A ). A similar argument shows that [id A ] = α ◦ [ λ ]. 10 / 11

. Main Theorem (H, Fima–Germain, ’15) . If A 1 and A 2 are separable, then the canonical surjection λ : A → A implements a KK-equivalence. . For ( A , E ) = ( A 1 , E 1 ) ∗ D ( A 2 , E 2 ), we have E A k : A → A k . Let ( X , ϕ X , ξ 0 ), ( Y k , ϕ k , η k ) be the GNS-repns for E and E k , resp. Consider the A - A C ∗ -correspondences ⊕ ( H 0 , π 0 ) := ( Y k ⊗ A k A , ϕ k ⊗ 1) “Γ / Γ 1 ⊔ Γ / Γ 2 ” k =1 , 2 ( H 1 , π 1 ) := ( X ⊗ D A , ϕ X ⊗ 1) “Γ / Λ” We then find S : H 0 → H 1 such that ( [ S ∗ ]) H 0 ⊕ H 1 , π 0 ⊕ π 1 , S implements α ∈ KK ( A , A ). A similar argument shows that [id A ] = α ◦ [ λ ]. Also, one can show that [id A ] = [ λ ] ◦ α . 10 / 11

. Theorem (Thomsen (2003)) . Let i k : D ֒ → A k and j k : A k ֒ → A be inclusion maps. Then, we have the exact sequence sequence for any separable B: ( i 1 ∗ , i 2 ∗ ) j 1 ∗ − j 2 ∗ K 0 ( D ) − − − − → K 0 ( A 1 ) ⊕ K 0 ( A 2 ) − − − − → K 0 ( A ) �     � ( i 1 ∗ , i 2 ∗ ) j 1 ∗ − j 2 ∗ K 1 ( A ) ← − − − − K 1 ( A 1 ) ⊕ K 1 ( A 2 ) ← − − − − K 1 ( D ) . 11 / 11