2019/06/20 by Schötz, Matthias
#06F25 (secondary) #47L60 (primary) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1906.08752
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 σ-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 σ-bounded, then the order of V is induced by the extremal positive linear functionals on V.