Gelfand-Naimark Theorems for Ordered *-Algebras

TL;DR

This paper extends the Gelfand-Naimark theorems to certain ordered *-algebras, including unbounded operator algebras, by constructing faithful representations as operators or functions under specific conditions.

Contribution

It generalizes classical representation theorems to broader classes of *-algebras, introducing new results for $\sigma$-bounded ordered vector spaces.

Findings

Faithful operator representations for $\sigma$-bounded ordered *-algebras

Representation as complex-valued functions under regularity conditions

Order structure induced by extremal positive linear functionals

Abstract

The classical Gelfand--Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for $\sigma$-bounded closed ordered *-algebras a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand--Naimark representation theorems to classes of *-algebras larger than C*-algebras, and which especially contain *-algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma$-bounded, then the order of V is induced by the extremal positive linear functionals on V.