Metric semi-groups of normal states determine F-algebras

Abstract

Abstract Let A be an F-algebra and S(A*) be the metric semi-group of normal states of the W*-algebra A*. We show that S(A*) determines A, when A* is of type I. In particular, when A is either the Fourier algebra A(G) or the Fourier–Stieltjes algebra B(G) of a type I locally compact group G, the metric semi-group S(A*) determines G up to opposition. A remark for the general case will also be given.