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Abstract
We study Köthe PDF-algebras. Using two (different yet natural) definitions of multiplication we obtain a wide class of natural algebras with either discontinuous or continuous multiplication. In this last case, we are able to fully characterize amenable Köthe PDF-algebras in terms of the defining Köthe matrix. This characterization shows an interesting and unexpected relation between algebraic and topological structures of amenable Köthe PDF-algebras.
Highlights
The notion of an amenable algebra goes back to the breakthrough paper of Johnson [14] where he proved that a locally compact group G is amenable, i.e. there exists an invariant mean on L∞(G) if and only if the convolution algebra L1(G) has the following property: every continuous derivation into any dual L1(G)-bimodule is inner
Piszczek characterization of amenable Banach algebras in terms of the exactness of a specific functor acting on admissible sequences
The aim of this paper is to study amenability properties of topological algebras given by countable projective limits of DF-spaces
Summary
The notion of an amenable algebra goes back to the breakthrough paper of Johnson [14] where he proved that a locally compact group G is amenable, i.e. there exists an invariant mean on L∞(G) if and only if the convolution algebra L1(G) has the following property: every continuous derivation into any dual L1(G)-bimodule is inner. This property served as a definition of an amenable Banach algebra and the paper [14] started investigation of the properties of amenable Banach algebras. Johnson and (independently) Helemskiĭ and his collaborators rephrased the amenability of an algebra A in terms of vanishing of the first continuous Hochschild cohomology group of A with coefficients in an arbitrary dual Banach A-bimodule. Helemskiĭ [12] launched the functorial machinery to come up with the Communicated by Zinaida Lykova
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