# Banach synaptic algebras

## TL;DR

This paper characterizes Banach synaptic algebras as those isomorphic to self-adjoint parts of Rickart C*-algebras and explores their relation to AW*-algebras and generalized Hermitian algebras.

## Contribution

It provides a characterization of Banach synaptic algebras using representation theorems and establishes their connection to well-known operator algebra structures.

## Key findings

- Banach synaptic algebras are isomorphic to self-adjoint parts of Rickart C*-algebras.
- Conditions are given for a Banach synaptic algebra to be isomorphic to an AW*-algebra.
- Relationships between synaptic algebras and generalized Hermitian algebras are analyzed.

## Abstract

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Stormer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C*-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW*-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.