Projectivity of L^p(VN(G)) as a left A(G)-module
by
Hun Hee Lee
University of Waterloo
Coauthors: Brian Forrest and Ebrahim Samei

For any locally compact group G we can understand L^p(G) as a L¹(G)(the convolution algebra)-module under the convolution product. Dales and Polyakov (2004) proved that, for finite p which is greater than 1, L^p(G) is projective as a left L¹(G)-module if and only if G is compact. In this talk we focus the dual situation, namely a natural A(G)(the Fourier algebra)-module structure on L^p(VN(G)). We will show that, for finite p which is greater than 1, L^p(VN(G)) is a projective left operator A(G)-module when G is discrete and amenable. Conversely, we can show that, for finite p which is greater or equal to 2, L^p(VN(G)) is not projective when G is non-discrete group with approximation property. Moreover, the above module structure can be defined similarly in the case of Kac algebras and locally compact quantum groups, and some results about projectivity still hold.

Date received: April 25, 2008