Classification of unitary elements of a C*-algebra

Abstract: The Cuntz semigroup has emerged as an essential tool for the classification of (non-simple) C*-algebras. For instance, it has been shown that the functor Cu classifies positive elements of any C*-algebra of stable rank 1 (up to approximately unitarily equivalence). An immediate corollary is that the Cuntz semigroup is a complete invariant for AI algebras. In this talk, I will raise the question of classification of unitary elements of a C*-algebra (of stable rank 1). It is unlikely that the Cuntz semigroup alone is sufficient to classify these elements and one can speculate that an ingredient with $K_1$ flavor has to be added. Nevertheless, I will prove that this remains true when restricting to AF algebras and I will discuss how one could to extend this classification result to a larger class of C*-algebra.

Even though I will recall definitions of the Cuntz semigroup and classifying functor, it might good to point out that knowledge about C*-algebras are needed.

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 sju.webex.com/meet/nikolaei (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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