Ordered *-algebras (i.e. unital *-algebras endowed with a quadratic module of "positive" elements) provide an abstraction of *-algebras of possibly unbounded operators on Hilbert, or pre-Hilbert, spaces. In this sense, they are generalizations of C*-algebras to the unbounded setting. Closed ordered *-algebras, i.e. ordered *-algebras that are Archimedean as ordered vector spaces, are of special interest. In this talk, I will give an introduction to this concept of ordered *-algebras, I will show how pre-C*-algebras are equivalent to "uniformly bounded" closed ordered *-algebras, and then discuss the generalization of the Gelfand-Naimark representation theorems from C*-algebras, or pre-C*-algebras, to "σ-bounded" closed ordered *-algebras, i.e. *-algebras for which it is required that there exists a cofinal sequence (an assumption that holds, in particular, in the uniformly bounded case and in the countably generated case).
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