Analysis and Probability seminar

Abstract: Operator systems have a central role in the theory of operator algebras. Lately, the community has been actively considering selfadjoint operator subspaces, but which need not be unital. Those appear naturally, for example when comparing operator systems up to their stabilisations as in the work of Connes and van Suijlekom. The absence of a unit disrupts the link between the norm and the matrix order structure, creating significant difficulties in obtaining even fundamental theorems like the celebrated Arveson's Extension Theorem.

In the first part of this talk I will present how such problems are resolved by working with morphisms that their unitisation (in the sense of Werner) is completely isometric. Furthermore I will provide (old and new) characterisations of such maps in terms of extensions and gauge isometries in the sense of Russell), with a focus on approximately positively generated or singly generated spaces.

Stabilisations can be seen as an incarnation of Morita equivalence for operator systems, similar to Rieffel's notion for C*-algebras introduced in the 60s. In the second part of this talk I will give equivalent characterizations via Morita contexts, bihomomoprhisms and a symmetrisation product, while I will highlight properties that are invariant. Time permitted I will provide applications to rigid systems, function systems and non-commutative graphs.

This talk is based on joint works with Alexandros Chatzinikolaou, Joseph Dessi, George Eleftherakis, Se-Jin Kim, Apollonas Paraskevas, and Ivan Todorov.