An isomorphism theorem for infinite reduced free products

TL;DR

This paper proves an isomorphism theorem for infinite reduced free products of certain C*-algebras, showing they are isomorphic to the building blocks under specific conditions, extending understanding of free product structures.

Contribution

It establishes an isomorphism between infinite reduced free products of specific C*-algebras and the algebras themselves, generalizing previous results and including new classes of spaces.

Findings

Infinite reduced free products of certain C*-algebras are isomorphic to the original algebra.

The result applies to C([0,1]) with Lebesgue measure and the Jiang-Su algebra.

Under additional conditions, the theorem extends to C(X) for any contractible space.

Abstract

Let C be a separable unital C*-algebra, not isomorphic to the complex numbers, equipped with a faithful tracial state. 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 copies of C. Then the reduced free product A*D is isomorphic to D. If D has real rank zero and C is exact, 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.