An isomorphism theorem for infinite reduced free products

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.

💡 Research Summary

The paper establishes a striking isomorphism theorem for infinite reduced free products of C*-algebras. Let C be a separable unital C*-algebra (different from ℂ) equipped with a faithful tracial state, and let D denote the infinite reduced free product of countably many copies of C, written D = C∗ₙ∞. The main result shows that, under suitable hypotheses on a second algebra A, the reduced free product A∗ₙ∞ D is actually isomorphic to D itself.

Two families of hypotheses are considered. In the first, A is a unital direct limit of one‑dimensional NCCW complexes, again equipped with a faithful trace. Crucially, there must exist a unital, trace‑preserving embedding of A into the Jiang‑Su algebra ℤ that induces an isomorphism on K‑theory (both K₀ and K₁). Under these conditions the authors prove A∗ₙ∞ D ≅ D. In the second family, the exactness of C and the real‑rank‑zero property of D are assumed; then A may be taken to be C(X) for any contractible compact metric space X with any faithful probability measure. The same conclusion holds provided the image of K₀(C) under the trace is dense in ℝ and D∗ₙ∞ is exact.

The proof proceeds in two major stages. First, an “approximate intertwining” theorem (Theorem 2.2) is developed for direct systems of separable C*-algebras. Given two inductive sequences (Aₙ) and (Bₙ) together with finite subsets and maps μₙ:Aₙ→Bₙ, νₙ:Bₙ→Aₙ₊₁ satisfying a hierarchy of ε‑approximations, the theorem guarantees a unique limiting homomorphism μ:A→B that is in fact an isomorphism. This framework adapts the classic Elliott intertwining argument to the setting of reduced free products, where the algebras involved are not nuclear.

Second, the authors invoke recent uniqueness theorems for homomorphisms from one‑dimensional NCCW complexes (or from C(X) when X is contractible) into target algebras possessing stable rank one, real rank zero, weakly unperforated K₀, and a finite tracial simplex. Such results (cited from