Apollonas Paraskevas: [WS1] Embeddings and Extension Theory of Selfadjoint Operator Spaces

Speaker
Apollonas Paraskevas, University of Athens

Abstract
Operator systems play 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. In this talk we focus on
Werner’s unitisation of such spaces and on embeddings between them, that
is, completely isometric complete order maps whose unitisation remains
completely isometric. The notion of embeddings is closely related to gauge
maximal inclusions of Russell and to having an Arveson’s Extension Theorem
for completely contractive completely positive maps. We give a
characterisation of embeddings and several intrinsic conditions of the
selfadjoint operator space that allows various types of extensions of
maps. We focus on two classes that have tractable cone structure:
(approximately) positively generated and singly generated selfadjoint
operator spaces. Finally, we give some applications on extremal theory.

Joint work with Alexandros Chatzinikolaou, Evgenios Kakariadis and Se-Jin
Kim.