**ABSTRACT:**

We define new symbol classes for pseudodifferential operators and investigate their

calculus. The symbol classes are parametrized by

commutative convolution algebras. To every solid

convolution algebra over a lattice we associate a symbol

class that is reminiscent of a modulation space.

Then every operator with a symbol in such a class is

almost diagonal with respect to special wave packets (coherent states

or Gabor frames) and rate of almost diagonalization is described

precisely by the underlying convolution algebra.

Furthermore, the corresponding class of pseudodifferential operators is a

Banach algebra of bounded operators on $L^2(\Bbb R^d)$. If a version of

Wiener's lemma holds for the underlying convolution algebra, then

the algebra of pseudodifferential operators is closed under inversion.

The theory contains as a special case the fundamental results about

Sj\"ostrand's class and yields a new proof of a theorem of Beals.

This is a joint work with Karlheinz Gr\"ochenig.