Strictly Convex Banach Algebras

David Yost

doi:10.3390/axioms10030221

Abstract

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C∗-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.

Highlights

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Banach algebras are a class of objects with three structures: a vector space structure, a metrizable topology defined by a norm, and a multiplication
The Banach space axioms regulate the interplay between the norm and the vector space structure; in particular, the norm is positively homogeneous, and the metric it gives rise to is complete

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Banach algebras are a class of objects with three structures: a vector space structure, a metrizable topology defined by a norm, and a multiplication. A Banach algebra is called unital if it has an identity element for multiplication. The unit sphere of a Banach space is the collection of all elements with norm one.

Results

Conclusion