A Fraïssé theoretic approach to the Jiang--Su algebra

Abstract

In this paper, we give a Fraïssé theoretic proof of the result of X. Jiang and H. Su that the Jiang-Su algebra is the unique monotracial simple C*-algebra among all inductive limits of prime dimension drop algebras. The proof presented here is self-contained and quite elementary, and does not depend on any K-theoretic technology. We also partially recover the fact that every unital endomorphism of the Jiang-Su algebra is approximately inner.

Stably projectionless Fraïssé limits

“…Notably, the authors of [7] showed that the Jiang-Su algebra Z and the UHF algebras of infinite type are Fraïssé limits of suitable Fraïssé classes. Masumoto, in [24,25] and [23], obtained the same results with a 'by hand' approach, not relying on any classification theory.…”

“…Theorems 3.5 and 3.13 in [23] showed that K Z is a Fraïssé class and that the Jiang-Su algebra Z is its limit. Masumoto then analyzed the structure of K Zadmissible embeddings of Z into itself.…”

Stably projectionless Fraïssé limits

“…Notably, the authors of [6] showed that the Jiang-Su algebra Z and the UHF algebras of infinite type are Fraïssé limits of suitable Fraïssé classes. Masumoto [23,24,22] obtained the same results with a 'by hand' approach, not relying on any classification theory. Ghasemi [11] further analysed the connections between Fraïssé theory and strongly self-absorbing C * -algebras to give a self-contained and rather elementary proof for the well known fact that Z is strongly self-absorbing.…”

“…Theorems 3.5 and 3.13 in [22] showed that K Z is a Fraïssé class and that the Jiang-Su algebra Z is its limit. Masumoto then analysed the structure of K Z -admissible embeddings of Z into itself.…”