The Hochschild Cohomology of Roe Type Algebras

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Presentation Abstract

Presentation Date

11-21-2022

Abstract

In order to help us better understand the structure of a space we look for invariants, not only of the space but also invariants of its algebra of functions. One such invariant is the Hochschild (co)homology. Using the Hochschild-Kostant-Rosenberg theorem (for sufficiently well behaved commutative algebras) one may identify the Hochschild homology with differential forms and the cohomology with multivector fields. Thus, for a noncommutative algebra we may consider its Hochschild (co)homology as noncommutative analogs of differential forms and multivector fields respectively.

Many times in analysis we focus on the "small scale" structure of a metric space, e.g. continuity, derivations, etc. However, to examine the "large scale" structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we again look at invariants, and one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X,d_X) is coarsely equivalent to (Y, d_Y) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely topological definitions exist). We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules. Time permitting, we will briefly discuss how our methods might be generalized to crossed product C*-algebras and C*-Algebras generated by groupoids. For this talk we will not assume any prior knowledge of C*-algebras or (co)homology.

Additional Details

November 21, 2022 at 3:00 p.m. Math 103

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