A type of GNS-construction for Banach algebras

TL;DR

This paper demonstrates that every Banach algebra can be represented on a specific Banach space, especially when contained in autoperiodic functionals that separate points, enabling embedding into bounded operators on a reflexive space.

Contribution

It introduces a GNS-like construction for Banach algebras, showing their representability on Banach spaces under certain conditions.

Findings

Banach algebras can be represented on Banach spaces.

Embedding into B(E) is possible for algebras with autoperiodic functionals.

Such embeddings are into reflexive Banach spaces.

Abstract

We show that every Banach algebra A admits a representation on a certain Banach space E. In particular, any Banach algebra A contained in autoperiodic functionals on A such that separate the points of A could be imbedded in B(E) for some reflexive Banach space E.