[Settfa] Tuesday 11/05, Tristan Bice

[kubis at math.cas.cz]

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 Tuesday 11th May, 10:30am 
 
  Place: Zoom: 

https://cesnet.zoom.us/j/93170324507?pwd=NEtINFlIUGZEQVZZMmVJOHRNNUlYQT09

 
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 Speaker:Tristan Bice, IM CAS
 Title: Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

 Abstract  

 The first and foremost fundamental result in C*-algebra theory was a duality proved by Gelfand in the 40's, namely that locallycompact Hausdorff spaces are dual to commutative C*-algebras(=Banach *-algebras satisfying the C*-norm condition||a*a||=||a||2). This inspired much of modern C*-algebra theory which is predicated on the idea that elements of a non-commutative C*-algebra are functions on some kind of "non-commutative space". Making this precise has proved quite difficult, although a number of Gelfand extensions have been developed over the years where non-commutative C*-algebras are indeed represented on certain non-commutative topological structures. For example, in the 60's, Dauns and Hofmann showed how to represent non-commutative C*-algebras on C*-bundles, i.e. bundles of (potentially simpler) C*-algebras over locally compact spaces. In a different direction, in work starting from the 80's, Kumjian and Renault showed how to represent Cartan pairs (consisting of aC
 *-algebra and a Cartan subalgebra) on complex line bundles over étale groupoids. In our talk, we will outline our recent unification of these results where C*-algebras with some additional Cartan-like structure are represented on Fell bundles over étale groupoids. Our construction is also in a sense more elementary than its predecessors, the points of our base groupoid being ultrafilters (with respect to a "domination relation" defined from the product) much like in classical topological dualities due to Stone, Wallman and Milgram from the 30's and 40's. In particular, this different approach allows us to also extend the functorial aspect of the original Gelfand duality, which had so far been lacking in its non-commutative extensions.


 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .