Almost Elementary C*-Dynamics and classifiability of crossed products

Joachim Zacharias  (University of Glasgow)

15:25-16:25,August 2,2023   Online: Zoom Meeting ID: 336 111 4671 Passcode: RCOA2023

Abstract:

Motivated by the Toms-Winter conjecture and Kerr’s notion of almost finiteness for actions of amenable discrete groups on compact metric spaces, we propose a generalisation of almost finiteness to actions of discrete groups on general C*-algebras which we call almost elementary. It turns out that our concept also applies to just C*-algebras (G={e}), where it is essentially equivalent to Z-stability (ie classifiability) in the simple case, to actions of amenable groups on commutative algebras, where it coincides with Kerr’s almost finiteness, and also to actions of non-amenable groups on general C*-algebras, thereby unifying and extending various concepts. Our starting point is a generalisation of Kerr's notion of a castle, which for us is a simultaneous approximation of the algebra and the action, up to an arbitrarily small remainder, small in various dynamically tracial senses. It turns out that many different natural smallness conditions are all equivalent. In the case of no group action (G={e}) our condition is a weak form of being tracially AF or having tracial nuclear dimension 0, equivalent to Z-stability, for separable simple nuclear algebras, thus providing another equivalent condition to the Toms-Winter conjecture. Moreover, almost elementary actions lead to Z-stable crossed products, in line with it being a kind of dynamical Z-stability. There is also a connection with tracial oscillation zero.Joint with Joan Bosa, Francesc Perera and Jianchao Wu.

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