Fraïssé limits in functional analysis

Abstract

We provide a unified approach to Fraïssé limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We obtain in this general framework many known and new results about the Gurarij space and the Poulsen simplex, and at the same time establish their noncommutative analogs. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota.2000 Mathematics Subject Classification. Primary 46L07, 46A55; Secondary 46L89, 03C30, 03C98.

Approximate Injectivity

“…This theory applies to Banach spaces, too, because the category Ban of (real or complex) Banach spaces and their linear operators of norm at most 1 is locally ℵ 1 -presentable (see [2], 1.48). But in this category there is another -and probably more important-concept, of so-called approximate injectivity (see [15]), which is based on the fact that Ban is enriched over metric spaces. The basic idea is to replace the commutativity of diagrams by their "commutativity up to ε".…”

Embeddings and -Envelopes of Exact Operator systems

“…In [13], Kirchberg and Wassermann gave an example of a universal separable exact operator system S with nonexact C * -envelope. Another interesting example was recently constructed by Lupini in [14], namely, the Gurarij operator system GS, which is exact but does not admit any complete order embedding into any exact C * -algebra. Thus, in general, unlike C * -algebras, separable exact operator systems need not embed into O 2 .…”

The Lelek fan and the Poulsen simplex as Fraïssé limits

“…The space of real-valued continuous affine functions on the Poulsen simplex, from which one can recover the Poulsen simplex as the space of states, was constructed in a Fraïssé-theoretic framework by Conley and Törnquist (unpublished). Independently from our work, constructions of the Poulsen simplex as a Fraïssé limit were recently given in the work of Bartošová, Lopez-Abad, and Mbombo, and in the paper by Lupini [15], who uses the framework of the Fraïssé theory for metric structures in the sense of Ben Yaacov [2], and for that he works with the category of real-valued continuous affine functions on Choquet simplices which forms a category dual to the one of Choquet simplices. In fact, Lupini shows how to construct a number of objects from functional analysis as Fraïssé limits.…”