REPRESENTATIONS OF p-CONVOLUTION ALGEBRAS ON Lq-SPACES

For a nontrivial locally compact group G, consider the Banach algebras of p-pseudofunctions, p-pseudomeasures, p-convolvers, and the full group L-p-operator algebra. We show that these Banach algebras are operator algebras if and only if p = 2. More generally, we show that for q is an element of[1, infinity), if any of these Banach algebras can be represented on an L-q-space, then one of the following holds: (a) p = 2 and G is abelian; or (b) vertical bar 1/p - 1/2 vertical bar = vertical bar 1/q - 1/2 vertical bar. This result can be interpreted as follows: for p, q is an element of[1, infinity), the L-p- and L-q-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct p, q is an element of[1, infinity), if the L-p- and L-q-crossed products of a topological dynamical system are isomorphic, then 1/p + 1/q = 1. In order to prove this, we study the following relevant aspects of L-p-crossed products: existence of approximate identities, duality with respect to p, and existence of canonical isometric maps from group algebras into their multiplier algebras.