An isomorphism theorem for infinite reduced free products

Abstract

Let C_1, C_2, ... be a sequence of separable unital C*-algebras, equipped with faithful tracial states and satisfying a mild condition. Let A be a unital direct limit of one dimensional NCCW complexes, also equipped with a faithful tracial state. Suppose there is a unital trace preserving embedding of A in the Jiang-Su algebra which is an isomorphism on K-theory. (For example, A could be C([0,1]) with Lebesgue measure, or the Jiang-Su algebra itself.) Let D be the infinite reduced free product of the algebras C_n. Then the reduced free product A*D is isomorphic to D. If D is exact and the factors satisfy a blockwise real rank zero condition, then in place of A we can use C(X) for any contractible compact metric space X and any faithful tracial state on C(X). An example consequence is that the reduced free product of infinitely many copies of C([0,1]), with Lebesgue measure, is isomorphic to the reduced free product of infinitely many copies of the Jiang-Su algebra.

Improved the results to cover infinite products instead of infinite powers, expanded the open problem section, corrected minor errors and misprints. 37 pages