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Actions of unitary tensor categories on C*-algebras
Miércoles 17 Marzo 2021, 01:00pm
Accesos : 539
Contacto Carlos Segovia

Seminario de Categorías

Expositor: Roberto Hernández, Ohio State University

Resumen: A subfactor is a unital inclusion of simple von Neumann algebras/factors $Asubset B,$ and we study it via its standard invariant $cC,$ which corresponds to a unitary tensor category (UTC). We will review some subfactor reconstruction techniques by Popa, and Guionnet-Jones-Shlyakhtenko (GJS), highlighting that subfactors have quantum symmetries which are encoded by UTC-actions. Namely, we reinterpret the inclusion $Asubset B$ as encoding an action of its standard invariant $cC$ on $A,$ and reconstruct the overfactor $B$ as a generalized crossed-product by this UTC-action.

Large scale work of many researchers worldwide has recently culminated in the classification of C*-algebras, which is now at the level of Connes' classification of injective factors. Nowadays, C*-algebras is at a similar state to that of von Neumann algebras after Jones introduced the index for subfactors in the early 80s. Thereafter, great interest has arisen in constructing and classifying UTC-actions on C*-algebras, aiming to understand their structure from the viewpoint of quantum symmetries.

We will see that every UTC $cC$ acts on some simple, unital separable and monotracial C*-algebra constructed only from $cC$ by adapting diagrammatic and free probabilistic techniques from GJS. Using a 'Hilbertification' technique, we recover the UTC-action constructed by Brothier-Harglass-Penneys of $cC$ on the free group factor $Lmathbb{F}_infty.$ This is joint work with Hartglass. Finally, we will review some recent developments and obstructions to the existence of UTC actions on (classifiable) C*-algebras.

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https://www.youtube.com/watch?v=c5ZztPNPHBk

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