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.