Functional calculus and *-regularity of a class of Banach algebras

Abstract

Suppose that ( A , G , α ) (A,G,\alpha ) is a C ∗ C^* -dynamical system such that G G is of polynomial growth. If A A is finite dimensional, we show that any element in K ( G ; A ) K(G;A) has slow growth and that L 1 ( G , A ) L^1(G, A) is ∗ * -regular. Furthermore, if G G is discrete and π \pi is a “nice representation” of A A , we define a new Banach ∗ * -algebra l π 1 ( G , A ) l^1_{\pi }(G, A) which coincides with l 1 ( G ; A ) l^1(G;A) when A A is finite dimensional. We also show that any element in K ( G ; A ) K(G;A) has slow growth and l π 1 ( G , A ) l^1_{\pi }(G, A) is ∗ * -regular.

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

“…By [7, Prop. 1], see also [21,Prop. 1.3], we need to show that ρ(f ) ≤ π(f ) holds for all f ∈ A whenever π, ρ are * -representations of A satisfying kerπ ⊂ kerρ.…”