A simple, monotracial, stably projectionless C*-algebra

TL;DR

This paper constructs a unique simple, nuclear, stably projectionless C*-algebra with trivial K-theory and a single trace, exploring its properties and potential role in classifying similar algebras.

Contribution

It introduces a new monotracial, stably projectionless C*-algebra W and analyzes its structural properties and potential classification significance.

Findings

W has trivial K-theory and a unique tracial state.

Every nondegenerate endomorphism of W is approximately inner.

W may tensorially absorb itself, W⊗W ≅ W.

Abstract

We construct a simple, nuclear, stably projectionless C*-algebra W which has trivial K-theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C*-algebras as an analogue of the Cuntz algebra O_2. In this context, we show that every nondegenerate endomorphism of W is approximately inner and we construct a trace-preserving embedding of W into the central sequences algebra M(W)_\infty \cap W'. We conjecture that W\otimes W is isomorphic to W and we note some implications of this, for example that W would be absorbed tensorially by a certain class of nuclear, stably projectionless C*-algebras. Finally, we explain why W may play some role in the classification of such algebras.