Extending traces on fixed point [formula omitted] algebras under Xerox product type actions of compact Lie groups

Abstract

Let G denote a compact connected Lie group, with maximal torus T (of dimension d, say). Let n: G--t M,,C be a unitary representation, and form the nJL UHF C* algebra A = 0 M,,C (n is fixed); this is the infinite tensor product of copies of the n On matrix ring. We have an action tt: G 3 Aut(A), by setting a(g) = 0 Ad n(g), B. M. Baker suggested the name Xerox product type action, because the representation 7c is duplicated over and over. We may form the crossed product A x 1 G and the fixed point algebra A”” (or simply AC if there is little likelihood of ambiguity). By restriction of c( to T, we also obtain a Xerox action of the d-torus T (also called CI). Clearly AC is a unital subalgebra of A ‘. The principal result asserts that every trace on AC extends to a trace on A ‘. In the course of the proof, we also show that the Grothendieck group of A”, K,(AG), is finitely generated as a ring, and that KO(AT) (which is also a ring) is finitely generated as a K,,( A’)-module. There are several consequences of these results. The space of faithful pure traces on AG is a dense open subset of the pure trace space of A’, and moreover is homeomorphic to (Rd)+ +/W, the orbit space of the strictly positive d-tuples under the natural action of the Weyl group, provided that o! is faithful. If B= @FL, M,,,,C is a UHF algebra with corresponding product type action (not necessarily Xerox) /? = 0 Ad nrr then the natural inclusion A ‘3’ -+ (A @ B)G301 @ B induces a bijection on faithful pure traces (if c( is faithful), and the set of the latter is dense in the pure trace space of (A @B)‘,“@? Another result (not proved here) using the main result of the article asserts that a difference of characters of G divides an actual character (within the representation ring of G) if and only if its restriction to T divides a character of T.