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An isomorphism theorem for infinite reduced free products

Ilan Hirshberg, N. Christopher Phillips

TL;DR

This work investigates isomorphisms for infinite reduced free products in the nonnuclear C*-algebra setting. The authors develop an Elliott-type intertwining framework tailored to direct limits of one-dimensional NCCW complexes and contractible function algebras, leveraging trace embeddings into the Jiang–Su algebra $Z$ and detailed K-theory computations. Central results show that $A *_{r} C^{*_{r} extinfty} o C^{*_{r} extinfty}$ under two regimes, yielding concrete identifications such as $C([0,1])^{*_{r} extinfty} o Z^{*_{r} extinfty}$ and related contractible-space cases. These findings illuminate isomorphism phenomena for infinite free products beyond nuclear, $ ext{Z}$-stable regimes and raise questions about finite free products, nonnuclear extensions, and state-preserving rigidity in free-product constructions.

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.

An isomorphism theorem for infinite reduced free products

TL;DR

This work investigates isomorphisms for infinite reduced free products in the nonnuclear C*-algebra setting. The authors develop an Elliott-type intertwining framework tailored to direct limits of one-dimensional NCCW complexes and contractible function algebras, leveraging trace embeddings into the Jiang–Su algebra and detailed K-theory computations. Central results show that under two regimes, yielding concrete identifications such as and related contractible-space cases. These findings illuminate isomorphism phenomena for infinite free products beyond nuclear, -stable regimes and raise questions about finite free products, nonnuclear extensions, and state-preserving rigidity in free-product constructions.

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.
Paper Structure (4 sections, 15 theorems, 119 equations)

This paper contains 4 sections, 15 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main theorems
  4. Open questions

Key Result

Lemma 2.1

Let $(A_1, \tau_1)$, $(A_2, \tau_2)$, $(B_1, \sigma_1)$, and $(B_2, \sigma_2)$ be unital C*-algebras with given states. Suppose that for $j=1, 2$ we have unital homomorphisms $\varphi_j \colon A_j \to B_j$ such that $\sigma_j \circ \varphi_j = \tau_j$. Then these homomorphisms uniquely define a homo

Theorems & Definitions (32)

  • Lemma 2.1
  • Theorem 2.2
  • proof : Proof of \ref{['thm_Elliott_intertwining']}
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • ...and 22 more