Baum-Connes, coactions, and the Tilde Problem

Abstract: Trouble with the Baum-Connes Conjecture (with coefficients) can in some way be blamed upon the existence of groups for which the reduced-crossed-product functor is not exact. The full crossed product is exact but doesn't fix the conjecture. Efforts to fix the conjecture have focused upon the ``minimal exact crossed product'', whose existence is known through abstract nonsense, but a construction remains elusive. Baum, Guentner, and Willett propose a candidate formed in part by tensoring with a fixed action. Our contribution to the [BGW] ``exotic crossed product'' program involves composing the full crossed product with coaction functors, hoping that the shift to coactions will add new insights. In particular, we replace the [BGW] candidate by tensoring with a fixed coaction. For a long time we had a hard time proving that our functor is exact. The ``natural'' approach involves embedding into ``tilde multiplier algebras'' (which I'll define in the talk). But we can't see how to prove that this gives an exact functor, and we call this the Tilde Problem. To get around this, we initially proved exactness of our coaction functor another --- extremely unsatisfying --- way: a long odyssey through equivariant C*-correspondences, ``natural'' Morita equivalence, crossed-product duality, and --- the final humiliation --- appealing to exactness of the [BGW] crossed-product functor itself, completely thwarting our goal of doing everything within the realm of coactions. Fortunately, we recently saw how to use our incomplete knowledge of the tilde functor to prove exactness of our coaction functor. This is joint work with Steve Kaliszewski and Magnus Landstad.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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.