Isomorphisms and stable isomorphisms of non-selfadjoint operator algebras

Abstract: In this talk we address isomorphisms and stable isomorphisms of various classes of operator algebras. We state and resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multi-variable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.

We also address stable isomorphism of operator algebras, in connection with a recent work of Dor-On, Eilers and Geffen. Among others we show that if $\mathcal{A}$ and $\mathcal{B}$ are operator algebras with diagonals isomorphic to $c_0$ and $\mathcal{K}$ are the compact operators, then $\mathcal{A}\otimes\mathcal{K}$ and $\mathcal{B}\otimes\mathcal{K}$ are isometrically isomorphic if and only if $\mathcal{A}$ and $\mathcal{B}$ are isometrically isomorphic. If the algebras $\mathcal{A}$ and $\mathcal{B}$ satisfy an extra analyticity condition, a similar result holds with $\mathcal{K}$ being replaced by any operator algebra containing the compact operators. Time permitting we will discuss other classes of operator algebras and their stable isomorphisms, including tensor algebras of multivariable dynamical systems.

The above results come from various projects with C. Ramsey, E. Kakariadis and X. Lin.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( video )

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Noncommutative geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click meet.google.com/zjd-ehrs-wtx (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.

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