Unit-free versions of the Vidav-Palmer theorem and of the Blecher-Ruan-Sinclair non-associative characterization of unital $C^*$-algebras

TL;DR

This paper develops unit-free versions of key theorems in operator algebra theory, extending their applicability to broader algebraic contexts and providing new characterizations of unital $C^*$-algebras.

Contribution

It introduces unit-free forms of the Vidav-Palmer theorems and applies them to establish a unit-free characterization of unital $C^*$-algebras.

Findings

Established unit-free Vidav-Palmer theorems for associative and non-associative cases.

Derived a unit-free version of the Blecher-Ruan-Sinclair characterization.

Extended the theoretical framework for analyzing unital $C^*$-algebras.

Abstract

We prove unit-free versions of both the associative and the non-associative Vidav-Palmer theorems. Then these results are applied to prove a unit-free version of the Blecher-Ruan-Sinclair non-associative characterization of unital $C^*$-algebras.