We study group representations on L^p spaces and the associated Banach algebras F^p(G) and F^p_r(G). These are, respectively, generalizations of the full group C*-algebra, and the reduced group C*-algebra (introduced by Herz in the 70's, and Phillips in 2012, respectively). While some classical results from group C*-algebras carry over to the L^p analogs (for instance, a characterization of amenability of G in terms of the canonical map from full to reduced), the algebras F^p(G) and F^p_r(G) tend to be very rigid objects when p is not equal to 2. The main result of the talk is that for p and q not equal to 2, and for locally compact groups G and H, there is a contractive isomorphism F^p_r(G) \cong F^q_r(H) if and only if G and H are isomorphic, and p and q are either equal or H\"older conjgate. When p=q=1, this was first obtained by Wendel in the 60's. Similar conclusions hold for the L^p analogs of the von Neumann algebra of a group: the p-pseudomeasures and the p-convolvers.
This is based on joint work with Hannes Thiel (University of Muenster).