Isomorphisms of Algebras of Convolution Operators

Journal article, 2022

For p, q E [1, oo) , we study the isomorphism problem for the p-and q-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras CVp(G) of p-convolvers and PMp(G) of p-pseudomeasures, for p # 2. More generally, we show that if CVp(G) is isometrically isomorphic to CVq(H) , with p, q # 2 , then G must be isomorphic to H and p and q are either equal or conjugate. This implies that there is no Lp-version of Connes' uniqueness of the hyperfinite II1-factor. Similar results apply to the algebra PFp(G) of p-pseudofunctions, gen-eralizing a classical result of Wendel. We also show that other Lp-rigidity results for groups can be easily recovered and extended using our main theorem.Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between PFp(G) and CVp(G): if any such algebra is reflexive and amenable, then G is finite.