Semisimplicity of some class of operator algebras on Banach space (Q2464695)

Let \(G\) be a locally compact Abelian group and let \({\mathbf T}= T(g)_{g\in G}\) be a representation of \(G\) by means of isometries on a Banach space. Define \(W_{{\mathbf T}}\) to be the closure with respect to the weak operator topology of the set \(\{\widehat f({\mathbf T}): f\in L^1(G)\}\), where \(\widehat f({\mathbf T})= \int_G f(g)T(g)\,dg\) is the Fourier transform of \(f\in L^1(G)\) with respect to the group \({\mathbf T}\). Then \(W_{{\mathbf T}}\) is a commutative Banach algebra. In this paper, the semisimplicity problem for such algebras is studied. The main result says that if the Arveson spectrum \(\text{sp}({\mathbf T})\) of \({\mathbf T}\) is scattered, then \(W_{{\mathbf T}}\) is semisimple. Some related problems are also discussed.