  We investigate pseudodifferential operators on arbitrary locally compact
abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce
a version of Sjoestrand's class. Pseudodifferential operators with such symbols
form a Banach algebra that is closed under inversion. Since "hard analysis"
techniques are not available on locally compact abelian groups, a new
time-frequency approach is used with the emphasis on modulation spaces, Gabor
frames, and Banach algebras of matrices. Sjoestrand's original results are thus
understood as a phenomenon of abstract harmonic analysis rather than "hard
analysis" and are proved in their natural context and generality.
