Eusebio Gardella

Banach algebras generated by an invertible isometry of an Lp-space

Abstract We provide a complete description of those Banach algebras that are generated by an invertible isometry of an L p -space together with its inverse. Examples include the algebra P F p ( Z ) of p-pseudofunctions on Z , the commutative C ⁎ -algebra C ( S 1 ) and all of its quotients, as well as uncountably many ‘exotic’ Banach algebras. We associate to each isometry of an L p -space a spectral invariant called ‘spectral configuration’, which contains considerably more information than its spectrum as an operator. It is shown that the spectral configuration describes the isometric isomorphism type of the Banach algebra that the isometry generates together with its inverse. It follows from our analysis that these algebras are semisimple. With the exception of P F p ( Z ) , they are all closed under continuous functional calculus, and their Gelfand transform is an isomorphism. As an application of our results, we show that Banach algebras that act on L 1 -spaces are not closed under quotients. This answers the case p = 1 of a question asked by Le Merdy 20 years ago.

Quotients of Banach algebras acting on Lp-spaces

We show that the class of Banach algebras that can be isometrically represented on an Lp-space, for p≠2, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of Lp-spaces.

Group Algebras Acting on \(L^p\)-Spaces

For

p

p\in [1,\infty )

L^q

p∈[1,∞) we study representations of a locally compact group

Nonclassifiability of UHF ^{}-operator algebras

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