Yemon Choi (Lancaster University)

Tuesday 17th May, 2016 16:00-17:00 Maths 522


There have been several attempts to find "dual versions" of the
classical notions of almost-periodic, or weakly almost-periodic,
functions on groups; ideally one would like these spaces to be
C*-subalgebras of the group von Neumann algebra.

Granirer has proposed definitions of $AP(\widehat{G})$ and
$WAP(\widehat{G})$ which would be valid for any non-abelian group G,
but it is not known if these spaces are always subalgebras of VN(G).
Runde has proposed a modified definition of ``completely almost
periodic'' functionals, and observed that if G is either connected or
amenable then his space $CAP(\widehat{G})$ is a subalgebra of VN(G).

In this talk I will give an overview of the relevant definitions and
sketch some of the results mentioned above, Then I will report on some
recent work in progress, which shows that $CAP(\widehat{G})$ is always
a subalgebra of VN(G). Moreover, if G is discrete, we can show that
$CAP(\widehat{G})$ coincides with the invariant part of the uniform
Roe algebra, and hence coincides with the reduced group C* algebra in
many cases by work of Zacharias. If time permits, I will try to
outline how a positive solution to certain slice map problems would
imply that $CAP(\widehat{G})$ always coincides with the reduced group
C* algebra.

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