Gelfand-Naimark Theorems for Ordered -Algebras

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 opera...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Author: Schötz, Matthias
Format: Article
Language:English
Subjects:
Algebra
Mathematical functions
Operators (mathematics)
Representations
Theorems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422