Representations of finite groups and Cuntz-Krieger algebras

Abstract

We investigate the structure of the C *-algebras O p constructed by Doplicher and Roberts from the intertwining operators between the tensor powers of a representation p of a compact group. We show that each Doplicher-Roberts algebra is isomorphic to a corner in the Cuntz-Krieger algebra OA of a {0,1}-matrix A -A p associated to p. When the group is finite, we can then use Cuntz's calculation of the if-theory of OA to compute K.(O P ).Doplicher and Roberts have recently developed a duality theory for compact subgroups of SU(n, C) in which the dual object consists of a simple C*-algebra OG and an endomorpbism of OG [3,4]. The construction of OG is based on the concrete representation p of G in SU(n,C) rather than the abstract group G, so we prefer to call it O p ; our work originated in an attempt to find out how the structure of O p depends on the choice of representation. To this end we have computed the .fif-theory of O p for finite G, by embedding it as a corner in a Cuntz-Krieger algebra OA , and using Cuntz's calculation of K*{OA) [1]-One conclusion is that different representations of the same finite group can give algebras which have quite different if-theory, and hence are not even stably isomorphic or Morita equivalent.The algebra O p is constructed from the spaces of intertwining operators between the different tensor powers p n of p, and its structure is determined by the decompositions of p n into irreducibles, and hence by the decompositions of TT ® p for •K (= G. The combinatorics of the situation can be summed up in a bipartite graph with G as vertices, and our main observation is that these combinatorics are similar to those involved in Cuntz and Krieger's construction of a C*-algebra OA from a {0,1}-matrix A. When G is compact, A is infinite, and there are technical problems in transferring this combinatorial similarity to the C *-algebra level; indeed, we need to appeal to both [2] and [3] to do it. For finite groups, we can prove directly that O p is a corner in OA , and the simplicity of O p therefore follows from [2] alone. We shall go as far as we can in full generality, since we are optimistic that one can extend the results of [1] to cover infinite A, and use them to compute K»(O P ) for compact G along similar lines.

On the K-Theory of Cuntz–Krieger Algebras

“…In [10] , it was shown that 0 P was isomorphic to a corner in a C * -algebra generated by an infinite Cuntz-Krieger family; now we know by Theorem 1 that this Cuntz-Krieger algebra GA P is simple, we have K # (G p ) = K * (@AP) , and we can use Theorem 3 to compute K * (Gp) . In fact, we can do better: we can identify Z°° with the representation ring $Z(G), and K* (G p The following definitions and results are extensions of those given in [5, p.253] , where it is claimed that their results for finite matrices carry over to the infinite case.…”

“…These C* -algebras are built from spaces of intertwiners between tensor powers of a given faithful representation p: G-> SU(ffl) , where G is a compact group and l<dim($C) <°o i We refer to [10] for further details of their construction. Decomposing the tensor powers of p into irreducible components yields a countable 0-1 matrix A p, which may be shown to be irreducible and row finite.…”

“…[10] , [12]) . We thank him for his help and interest, and particularly for suggesting to us that we should interpret our results in terms of the representation…”

The 𝐶*-algebras of infinite graphs

“…For finite graphs, these are precisely the Cuntz-Krieger algebras O A : given E, take A to be the edge matrix A E defined by A E (e, f ) = 1 if r(e) = s(f ), 0 if r(e) = s(f ); (1.2) conversely, given an I × I matrix A, form the graph E A with vertex set I and incidence matrix A, and then C * (E A ) is isomorphic to O A in a slightly nonobvious way (see [9,Theorem 3] or [13,Proposition 4.1]). This correspondence carries over to locally finite graphs (graphs in which vertices receive and emit finitely many edges), and the classical uniqueness and simplicity theorems for Cuntz-Krieger algebras have elegant extensions to these graph algebras [11,12].…”

“…We can then use the results of [7] to obtain uniqueness and simplicity theorems for C * (E). It is not clear whether the reverse passage from Cuntz-Krieger algebras to graph algebras is always possible for graphs which are not locally finite: the obvious analogue of the isomorphism of O A onto C * (E A ) given in [9] and [13] would involve infinite sums of the sort we have to avoid in abstract C * -algebras. We shall begin by giving precise definitions and stating our main results.…”