  In this paper we introduce a new family of topological convolution algebras
of the form $\bigcup_{p\in\mathbb N} L_2(S,\mu_p)$, where $S$ is a Borel
semi-group in a locally compact group $G$, which carries an inequality of the
type $\|f*g\|_p\le A_{p,q}\|f\|_q\|g\|_p$ for $p > q+d$ where $d$ pre-assigned,
and $A_{p,q}$ is a constant. We give a sufficient condition on the measures
$\mu_p$ for such an inequality to hold. We study the functional calculus and
the spectrum of the elements of these algebras, and present two examples, one
in the setting of non commutative stochastic distributions, and the other
related to Dirichlet series.
