Geodesic
Gelfand–Naimark theorems for ordered $^*$-algebras
Canadian journal of mathematics, Tome 75 (2023) no. 4, pp. 1272-1292

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DOI

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.
DOI : 10.4153/S0008414X22000359
Mots-clés : *-algebra, quadratic module, operator algebra, representation theory 
Schötz, Matthias. Gelfand–Naimark theorems for ordered $^*$-algebras. Canadian journal of mathematics, Tome 75 (2023) no. 4, pp. 1272-1292. doi: 10.4153/S0008414X22000359
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     title = {Gelfand{\textendash}Naimark theorems for ordered $^*$-algebras},
     journal = {Canadian journal of mathematics},
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     year = {2023},
     volume = {75},
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