This is an announcement for the paper "Banach algebras generated by an invertible isometry of an $L^p$-space" by Eusebio Gardella and Hannes Thiel.

Abstract: We study and classify Banach algebras that are generated by an invertible isometry of an $L^p$-space together with its inverse. We associate to each such isometry a spectral invariant which contains considerably more information than its spectrum as an operator. We show that this invariant describes the isometric isomorphism type of the Banach algebra that the isometry generates together with its inverse. In the case of invertible isometries with full spectrum, these Banach algebras parametrize all completions of the group algebra $\mathbb{C}[\mathbb{Z}]$ corresponding to unital, contractive representations on $L^p$-spaces. The extreme cases are the algebra of $p$-pseudofunctions on $\mathbb{Z}$, and the commutative $C^*$-algebra $C(S^1)$. Moreover, there are uncountably many non-isometrically isomorphic "intermediate" algebras, all of which are shown to be closed under continuous functional calculus.

Archive classification: math.FA math.OA

Mathematics Subject Classification: Primary: 46J40, 46H35. Secondary: 47L10

Remarks: 43 pages

Submitted from: gardella@uoregon.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1405.5589

or

http://arXiv.org/abs/1405.5589