Topological convolution algebras

In this paper we introduce a dual of reflexive Fréchet counterpart of Banach algebras of the form p∈NΦp' (where the Φp' are (dual of) Banach spaces with associated norms {dot operator}_p), which carry inequalities of the form ab_p≤A_p,qa_qb_p and ba_p≤A_p,qa_qb_p for p>q+d, where d is preassigned and A_p,q is a constant. We study the functional calculus and the spectrum of the elements of these algebras. We then focus on the particular case Φp'=L2(S,μp), where S is a Borel semi-group in a locally compact group G, and multiplication is convolution. We give a sufficient condition on the measures μ_p for such inequalities to hold. Finally we present three examples, one is the algebra of germs of holomorphic functions in zero, the second related to Dirichlet series and the third in the setting of non-commutative stochastic distributions.