 

Rufus Willett
Approximate ideal structures and Ktheory
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Published: 
December 28, 2020. 
Keywords: 
Kunneth formula, BaumConnes conjecture, controlled Ktheory, MayerVietoris sequence. 
Subject: 
46L80, 46L85. 


Abstract
We introduce a notion of approximate ideal structure for a C*algebra, and use it as a tool to study Ktheory groups. The notion is motivated by the classical MayerVietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of OyonoOyono and Yu in the setting of controlled Ktheory of filtered C*algebras; we do not, however, use that language in this paper.
We give two main applications. The first is a vanishing result for Ktheory that is relevant to the BaumConnes conjecture. The second is a permanence result for the Kunneth formula in C*algebra Ktheory: roughly, this says that if A can be decomposed into a pair of subalgebras (C,D) such that C, D, and C∩ D all satisfy the Kunneth formula, then A itself satisfies the Kunneth formula. 

Acknowledgements
This work was started during a sabbatical visit to the University of
Munster. I would like to thank the members of the mathematics department there for their warm hospitality.
I would like to particularly thank Clément Dell'Aiera, Dominik Enders, Sabrina Gemsa, Hervé OyonoOyono,
Ian Putnam, Aaron Tikuisis, Stuart White, Wilhelm Winter, and Guoliang Yu for numerous enlightening conversations relevant to the topics of this paper.
The support of the US NSF through grants DMS 1564281 and DMS 1901522 is gratefully acknowledged.
Finally, my thanks to the anonymous referee for a careful reading of the paper.


Author information
Rufus Willett:
Mathematics Department
University of Hawaii at Manoa
Keller 401A, 2565 McCarthy Mall
Honolulu, HI 96822, USA
rufus@math.hawaii.edu

