A Simple Separable Exact C*-Algebra not Anti-isomorphic to Itself

TL;DR

This paper constructs a specific exact, simple, separable C*-algebra that is not isomorphic to its opposite, with detailed properties and explicit K-theory calculations, expanding understanding of non-self-isomorphic algebras.

Contribution

It provides the first known example of a simple, separable, exact C*-algebra not isomorphic to its opposite, with comprehensive structural and K-theoretic analysis.

Findings

D is not isomorphic to its opposite algebra

K_0(D) = Z[1/3], K_1(D) = 0

D absorbs the Jiang-Su algebra Z and the 3^{inite} UHF algebra

Abstract

We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3^{\infty} UHF algebra. We can also explicitly compute the K-theory of D, namely K_0 (D) = Z[1/3] with the standard order, and K_1 (D) = 0, as well as the Cuntz semigroup of D.