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.

No abstract

No abstract

No abstract