On the K-Theory of Cuntz–Krieger Algebras

Abstract

We extend the uniqueness and simplicity results of Cuntz and Krieger to the countably infinite case, under a row-finite condition on the matrix A. Then we present a new approach to calculating the K-theory of the Cuntz-Krieger algebras, using the gauge action of T, which also works when A is a countably infinite 0-1 matrix. This calculation uses a dual Pimsner-Voiculescu six-term exact sequence for algebras carrying an action of T. Finally, we use these new results to calculate the K-theory of the Doplicher-Roberts algebras. §L Introduction In [4], [5], [3], Cuntz and Krieger studied the C* -algebras generated by a family of n non-zero partial isometries S, f satisfying the Cuntz-Krieger relationswhere A is an nX n, 0-1 matrix with no zero row or column. Let SA denote the set of finite sequences fi= (^i, In particular, if A is irreducible (i.e. for all i,/, there is a strictly positive integer m=m(i, /) such that A m (i, /) =£0) and is not a permutation matrix, then A satisfies condition (l) . It was shown in [5] that when A satisfies (l) , the C * -algebra generated by the Si, i=l,-~, n is independent of the choice of the partial isometries Si, and simple whenever A is irreducible. It may therefore be denoted by 6 A-More precisely, in [5, 2.13 and 2.14] it was shown that:• . ( i ) Suppose that A is a finite {0,1} matrix satisfying (l) and Si, T i, i=l,°--,n, are two families of non~ zero partial isometries satisfying the same Cuntz-Krieger relations (l) . Then the map S,-H * Ti extends to an isomorphism of C* (Si,-,S«) ontoC*(Ti,-,T»).( ii ) If the matrix A is irreducible and not a permutation matrix, then the C* -algebra 6 A = C* (Si,-,S W ) is simple.In the next section we give conditions (j) on a countably infinite 0-1 matrix A, under which the following theorem holds: Theorem 1, ( i ) Suppose that A is a countably infinite {0,1} matrix satisfying (j) and Si, Ti, i^N are two families of non-zero partial isometries satisfying the same infinite Cuntz-Krieger relations. Then the map Si h-» T, extends to an isomorphism of C * (S i) onto C * ( T ,-) . ( ii ) If the matrix A row-finite and irreducible then the C * -algebre &&=€* (S/ ) is simple.While condition (j) is analogous to condition ( I ), in order to get the simplicity result, we must assume irreducibility and a finiteness condition to ensure that certain topological obstacles do not occur. In the third section we review the proof of the following theorem of Kishimoto and Takai, [9, Theorem 2] , since we shall need explicit details of the isomorphism later. K-THEORY OF CUNTZ-KRIEGER ALGEBRAS 417Theorem 2 8 Let B be unital C * -algebra and $ a strongly continuous action of a compact group G which has large spectral subspaces, then the fixed point algebra B® is stably isomorphic to BX &G.In the fourth section, we calculate the K-theory of the GA defined in the first section, generalising the results for finite matrices (see [5] , [3] ) . In particular, we prove the following:

Cuntz-Krieger algebras of infinite graphs and matrices

“…However, the approximation technique based on Lemma 1.2 is substantially different from the one used by Pask and Raeburn in [25].…”

“…Every graph algebra and Cuntz-Krieger algebra carries a canonical gauge action γ of T. As in [25], we compute K-theory using the dual Pimsner-Voiculescu sequence for γ. In general, if α : T → Aut A is an action of T on a C * -algebra A, then the dual Pimsner-Voiculescu sequence is a six-term exact sequence…”

“…It is understood in principle how to adapt the general theory to cover graphs with sinks [1, §1], but calculating the K-theory requires some work. Our approach to the computation of K-theory is new: we use the skew-product graphs of Kumjian and Pask [19] to avoid much of the usual chasing through stable isomorphisms and duality [8], [25], [23]. Thus this approach should be of interest even to those who only encounter graphs without sinks.…”

On graphC*-algebras

“…Let P\{x) and P 2 (x) stand for P(x) (see (2) -A' r )T° and K x (0 r ) = Ker(l -A' r )T°( see [2,15]). In case of general f, see [7].…”

Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory

“…Not only will we find a new version of this index equality, but we will see that it allows us to use APS boundary conditions to obtain interesting index pairings, and consequences, that were previously unknown. For instance we show that the complicated K-theory calculations of [14] can be given a simple functorial description.…”