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The K-Theory of a Simple Separable Exact C*-Algebra Not Isomorphic to Its Opposite Algebra

N. Christopher Phillips, Maria Grazia Viola

Abstract

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C$^\ast$-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the $K_0$-group, the $K_1$-group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem. This is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in earlier work.

The K-Theory of a Simple Separable Exact C*-Algebra Not Isomorphic to Its Opposite Algebra

Abstract

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the -group, the -group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem. This is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in earlier work.

Paper Structure

This paper contains 7 sections, 24 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The continuous Rokhlin property and the Universal Coefficient Theorem
  4. The construction
  5. The main step
  6. C*-algebras not isomorphic to their opposite algebras
  7. Open questions

Key Result

Lemma 2.4

Let $A$ be a C*-algebra, and let $\tau$ be a tracial state on $A$. Then $\tau^{\mathrm{op}}$ is a tracial state on $A^{\mathrm{op}}$ and, as von Neumann algebras, we have $\pi_{\tau^{\mathrm{op}}} (A^{\mathrm{op}})" \cong [\pi_{\tau} (A)"]^{\mathrm{op}}$.

Theorems & Definitions (67)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 57 more