On the spectral theory of Rickart Ordered *-algebras

TL;DR

This paper introduces RO*-algebras, explores their properties, and proves a spectral theorem for self-adjoint elements, advancing the understanding of their structure and bounded elements.

Contribution

It defines RO*-algebras, shows bounded elements form a C*-normed set, and proves a spectral theorem for self-adjoint elements, extending spectral theory to this class.

Findings

Bounded elements in RO*-algebras form a C*-normed set.

The structure of commutative subalgebras is characterized.

A spectral theorem for self-adjoint elements is established.

Abstract

RO*-algebras are defined and studied. For RO*-algebra T, using properties of partial order, it is established that the set of bounded elements can be endowed with C*-norm. The structure of commutative subalgebras of T is considered and the Spectral Theorem for any self-adjoint element of T is proven.