The Jiang-Su algebra is strongly self-absorbing revisited

Abstract

We give a shorter proof of the fact that the Jiang-Su algebra is strongly self-absorbing. This is achieved by introducing and studying so-called unitarily suspended endomorphisms of generalized dimension drop algebras. Along the way we prove uniqueness and existence results for maps between dimension drop algebras and UHF-algebras.

Nonseparable CCR algebras

“…Apparently the simplest instance of Question 5.1 not resolved by our Theorem 1 is the case of p prime M p (C). Probably the most interesting case is when A is the Jiang-Su algebra Z, whose Elliott invariant is equal to that of the complex numbers (see [15], also [12] and [17] for most recent treatments).…”

Strongly self‐absorbing C∗‐algebras and Fraïssé limits

“…Thus, it would be desirable to find a proof of the fact that Z is strongly self-absorbing, which replaces the second step by arguments that do not depend (at least not so heavily) on the K-theoretic data and the classification tools. Towards this goal, recently a new and easier proof has been discovered by A. Schemaitat [18], which uses a characterization of Z as a stationary inductive limit of a generalized prime dimension-drop algebras and a trace-collapsing endomorphism, given in [17].…”

Strongly self-absorbing $C^*$-algebras and Fraïssé limits

“…Thus, it would be desirable to have a direct proof of the fact that Z is strongly self-absorbing, which does not depend (at least not so heavily) on the K-theoretic data and the classification tools. Towards this goal, very recently, a new and easier proof has been discovered by A. Schemaitat [18], which uses a characterization of Z as a stationary inductive limit of a generalized prime dimension-drop algebras and a trace-collapsing endomorphism, given in [17]. In this paper we give a different proof of the fact that Z is strongly self-absorbing, which follows a more general approach, and does not use any classification tools nor any characterization of Z.…”