Graded K-theory and Graph Algebras

Transcription

1 Alex Kumjian 1 David Pask 2 Aidan Sims 2 1 University of Nevada, Reno 2 University of Wollongong GPOTS, 1 June 2018 Miami University, Oxford

2 Introduction A C -algebra A endowed with an automorphism of order two is said to be Z 2 -graded or simply graded. Forty years ago Kasparov introduced KK, a bivariate K-functor for graded C -algebras, that revolutionized the study of K-theory for trivially graded C -algebras. The work [KPS] reported on here grew out of an attempt by David Pask, Aidan Sims and me to understand the K-theory of graded graph C -algebras. We proved graded versions of Pimsner s six-term exact sequences for KK-groups associated to Cuntz-Pimsner algebras and derived the existence of an exact sequence for computing the K-theory of graded graph C -algebras that generalizes the usual one for computing the K-theory of trivially graded graph C -algebras. We also proved a graded version of the Pimsner-Voiculescu six-term exact sequence.

3 Graded C -algebras A C -algebra A equipped with an automorphism α of order two is said to be graded and α is called the grading automorphism or simply the grading. Denote the induced grading on M(A) by α. An element a A is said to homogeneous of degree a = i where i Z 2 if α(a) = ( 1) a a. Every a A may be expressed uniquely as a sum a = a 0 + a 1 where a i = i. Thus A = A 0 A 1 where A i consists of all a A of degree i. We say that A is inner graded if there is a self-adjoint unitary u M(A) s.t. α(a) = uau for all a A. The first (complex) Clifford algebra is the C -algebra C 1 = C C endowed with the grading α(x, y) = (y, x). There is a graded tensor product A B of graded C -algebras s.t. if a, a A and b, b B are homogeneous elements (a b) = a + b, (a b)(a b ) = ( 1) b a aa bb. Note that C 2 := C 1 C 1 = M2 (C) with inner grading determined by u = diag (1, 1). More generally, C n+1 := C n C 1.

4 Graph C -algebras Graph C -algebras owe their origin to Cuntz algebras and more generally Cuntz-Krieger algebras. Let E = (E 1, E 0, r, s) be a directed graph with vertices E 0, edges E 1 and maps r, s : E 1 E 0. Suppose 0 < r 1 (v) < for all v. Let C (E) denote the universal C -algebra generated by an orthogonal set of projections {p v : v V } and a family of partial isometries {t e : e E} satisfying i t e t e = p s(e) for every e E 1, ii p v = r(e)=v t et e for every v E 0. Given a map δ : E 1 Z 2, the universality of C (E) ensures the existence of a grading α = α δ such that for all v E 0, e E 1 α(p v ) = p v and α(t e ) = ( 1) δ(e) t e. If δ(e) = 1 for all e E 1, α is called the standard grading.

5 Some examples Consider the graph B n with B 0 n := {v} and B 1 n := {e 1,..., e n }. B 2 Since B 1 has a one edge and one vertex the definition implies that C (B 1 ) = C(T). If n 2, C (B n ) = O n. Define the directed cycle C n with C 0 n = {v 1,..., v n } and C 1 n = {e 1,..., e n } such that r(e i ) = v i and s(e i ) = v i+1 if i < n and s(e n ) = v 1. Then C (B n ) = M n (C(T)). The center is generated by i t e i t en t e1 t ei 1. Define the graph Ω by Ω 0 = Ω 1 := N with s(n) = n + 1 and r(n) = n. Then C (Ω) = K(l 2 (N)).

6 Graded K-theory We use Kasparov s KK bifunctor (see [K], [B, 17.3]) to define the K-theory of a graded C -algebra. See [vd, H] for other approaches. Definition Let (A, α) be a σ-unital graded C -algebra. Define the graded K groups of A as follows: 0 (A, α) := KK(C, A), 1 (A, α) := KK(C, A C 1 ). When α is understood from context write i Note: We have (C) = (Z, 0) and If A is inner graded, then i (C 1 ) = (0, Z). (A) = K i (A). (A) for i (A, α). Skandalis has shown that under mild hypotheses the six-term exact sequence for extensions also holds for graded K-theory (see [S]).

7 The K-theory of graded graph C -algebras Let E be a graph s.t. 0 < r 1 (v) < for all v. Let δ : E 1 Z 2 be a map and let α = α δ. Let A δ E be the E 0 E 0 matrix defined by A δ E (v, w) = e ve 1 w ( 1) δ(e). Theorem The following sequence is exact. 0 1 (C (E), α) ZE 0 1 (Aδ E )t ZE 0 0 (C (E), α) 0 Example (a) Let C n be the directed cycle as above and let δ(e i ) = 1 for i = 1,..., n. By the above theorem, K gr (C (C n ), α) = (Z, Z) if n is even and K gr (C (C n ), α) = (Z 2, 0) if n is odd.

8 Example (b) Let B n be the graph with one vertex and n edges and let δ : Bn 1 {0, 1} be a map. Set p := δ 1 (1) and q := δ 1 (0). Note that C (B n ) = O n and A δ B n is a 1 1 matrix with entry q p. Therefore we have (cf. [H, 4.11]) { K gr (O n ) (Z 1+p q, 0) if 1 + p q 0, = (Z, Z) otherwise. Example (c) We define a graph E as follows: set E 0 := {v n : n Z} and E 1 := {e n, f n : n Z}; the range and source maps are given by r(e n ) = r(f n ) = v n and s(e n ) = v n+1 and s(f n ) = v n for n Z. Define δ : E 1 Z 2 by δ(e n ) = δ(f n ) = 1 for all n. Then A δ E (v m, v n ) = 1 if n = m, m + 1 and 0 otherwise; by a routine computation, 0 (C (E), α) = Z[ 1 2 ] and K gr 1 (C (E), α) = 0.

9 Graded Pimsner-Voiculescu sequence Let (A, α) be a σ-unital graded C -algebra and let γ Aut(A) s.t. γα = αγ. There are two natural gradings β k for k = 0, 1 on the crossed product A γ Z. Let j : A A γ Z denote the natural inclusion and let u M(A) be the canonical unitary such that j(γ(a)) = uj(a)u for all a A. Then for k = 0, 1 the gradings are determined by the formulae: β k (u) = ( 1) k u, β k (j(a)) = j(α(a)) for all a A. Theorem The following six-term exact sequence is exact for k = 0, 1. 0 (A, α) id ( α ) k γ j (A, α) 0 0 (A γ Z, β k ) 1 (A γ Z, β k ) gr 1 (A, α) K1 (A, α) j id ( α ) k γ

10 Example (d) Take A = C with the trivial grading and let γ be the identity automorphism. Then A γ Z = C (Z) = C(T) and β 0 is trivial but not β 1. Under the above isomomorphism β 1 (f )(z) = f ( z). In this case we have id ( α ) k γ = 2 id. Since 0 (A, α) = K 0(C) = Z and 1 (A, α) = K 1(C) = 0, we have Example (e) 0 (C(T), β1 ) = Z 2 and 0 (C(T), β1 ) = 0. Now let E and δ be as in Example (c) above and let γ be the automorphism of C (E) such that γ(p vn ) = p vn+1, γ(s en ) = s en+1, γ(s fn ) = s fn+1 for all n. Note that the map induced by the grading on i (C (E), α) is the identity (see [KPS, 8.8]) and the map induced by γ is multiplication by 2. So if we take the grading β 1 on A γ Z the map id ( α )γ on 0 (C (E), α) is a bijection. Hence i (A γ Z, β 1 ) = 0 for i = 0, 1.

11 [B] B. Blackadar, K-theory for operator algebras, [vd] A. van Daele, Graded K-theory for Banach algebras I, [H] U. Haag, On Z/2Z-graded KK-theory and its relation with the graded Ext-functor, [K] G.G. Kasparov, The operator C -functor and extensions of C -algebras, [KPS] Kumjian, Pask and Sims, Graded C -algebras, graded K-theory and twisted P-graph C -algebras, preprint. [P] M. Pimsner, A class of C -algebras generalizing both Cuntz-Krieger algebras and crossed products by Z, Fields Inst. Commun., 12, (Waterloo, ON, 1995), , Amer. Math. Soc., Providence, RI, [S] G. Skandalis, Exact sequences for the Kasparov groups of graded algebras, 1985.

12 Thanks! Questions?

13 An exact sequence for graded Cuntz-Pimsner algebras Let (A, α) be a σ-unital graded C -algebra and let X be a countably generated graded correspondence over A. There is an element [X ] KK(A, A) and a grading α X on the Cuntz-Pimsner algebra O X compatible with the map i : A O X. Since 0 (A) = KK(C, A), the product with [id A] [X ] yields an endomorphism on 0 (A) denoted A ([id A ] [X ]) (cf. [P, 4.9]). Theorem (Pimsner) With notation as above suppose that left action of A on X is injective and by compacts. Then the following is exact. 0 (A, α) A ([id A ] [X ]) i (A, α) 0 0 (O X, α X ) 1 (O X, α X ) gr 1 (A, α) K1 (A, α) i A ([id A ] [X ])

14 If there is time, here s one more example. Example (f) We define a graph E as follows: set E 0 := {v n : n Z} and E 1 := {e n, f n : n Z}; the range and source maps are given by r(e n ) = r(f n ) = v n and s(e n ) = s(f n ) = v n+1 for n Z. Define δ : E 1 Z 2 by δ(e n ) = 0 and δ(f n ) = 1 for all n. Then A δ E is zero and, hence, the map ZE 0 ZE 0 is the identity in the exact sequence above. Thus, i (C (E), α) = 0 for i = 0, 1. Note that C (E) is Morita equivalent to the UHF algebra M 2.