Anti-C*-algebras

TL;DR

This paper introduces anti-C*-algebras, a new class of Banach algebras, and explores their relationship with C*-ternary rings, including their structure, semisimplicity, and characterizations, addressing longstanding questions in operator algebra theory.

Contribution

It defines anti-C*-algebras, shows their role in the structure of C*-ternary rings, and provides new characterizations of TROs, advancing understanding in operator algebra theory.

Findings

Anti-C*-algebras are semisimple Banach algebras.

The normed embedding of C*-ternary rings decomposes into a C*-algebra and an anti-C*-algebra.

New characterizations of C*-ternary rings isomorphic to TROs are provided.

Abstract

We introduce a class of Banach algebras that we call anti-C*-algebras. We show that the normed standard embedding of a C*-ternary ring is the direct sum of a C*-algebra and an anti-C*-algebra. We prove that C*-ternary rings and anti-C*-algebras are semisimple. We give two new characterizations of C*-ternary rings which are isomorphic to a TRO (ternary ring of operators), providing answers to a query raised by Zettl in 1983, and we propose some problems for further study.