Gromov's translation algebras, growth and amenability of operator algebras

Translation algebras of finitely generated **-algebras of bounded linear operators on a separable Hilbert space are introduced. Two equivalent forms of amenability for finitely generated **-algebras in terms of the existence of Følner sequences are introduced. These are related to the existence of traces on the associated translation algebra and, in the context of C*C*-algebras, are related to weak-filterability and to the existence of hypertraces.