On a Generalized Fraïssé Limit Construction and Its Application to the Jiang–su Algebra

Abstract

In this paper, we present a version of Fraïssé theory for categories of metric structures. Using this version, we show that every UHF algebra can be recognized as a Fraïssé limit of a class of C*-algebras of matrix-valued continuous functions on cubes with distinguished traces. We also give an alternative proof of the fact that the Jiang–Su algebra is the unique simple monotracial C*-algebra among all the inductive limits of prime dimension drop algebras.

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.…”

“…The need of this technical restriction on 'allowed' morphisms from objects in K to Kstructures is due to the absence of the Hereditary Property. This absence is usually irrelevant in the discrete setting (for instance the classes considered by Irwin and Solecki in [18] do not have the Hereditary Property), but it creates technical issues in the continuous setting (see, e.g., the introduction of [25]).…”

“…Finally, §5 uses the previous sections to prove the NAP for our classes of interest, and contains the proof of our main result. Lastly, in Appendix A we deal with the issue of what kind of maps one obtains homogeneity for, hereby answering a question of Masumoto from [25].…”

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.…”

“…Finally, §5 uses the previous sections to prove NAP for our classes of interest, and contains the proof of our main result. In the Appendix we discuss for which maps one obtains homogeneity, answering a question of Masumoto from [24].…”