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Multiplications and involutions on vector spaces. (English) Zbl 0953.46023

The aim of this paper is to present
1) an abstract algebraic form of the classical Vidav-Palmer theorem, pertaining to a characterization of the existence of a \(C^*\)-algebra structure on a given unital Banach algebra and
2) an abstract algebraic formulation of the result of Bucy-Maltese, concerning the integral representation of a given positive linear form on a unital commutative Banach \(*\)-algebra.

MSC:

46H05 General theory of topological algebras
46K05 General theory of topological algebras with involution
46A55 Convex sets in topological linear spaces; Choquet theory

Citations:

Zbl 0930.00011
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References:

[1] F. F. Bonsall and J. Duncan,Complete Normed Algebras, Springer-Verlag, Berlin (1973). · Zbl 0271.46039
[2] R. S. Bucy and G. Maltese, ”A representation theorem for positive functionals on involutive algebras,”Math. Ann.,162, 364–367 (1966). · Zbl 0132.10804 · doi:10.1007/BF01369109
[3] A. Mallios,Topological Algebras. Selected Topics, North-Holland, Amsterdam (1986). · Zbl 0597.46046
[4] Y. Tsertos, ”Representations and extensions of positive functionals on *-algebras,”Boll. U. M. I.,8, 541–555 (1994). · Zbl 0834.46038
[5] Y. Tsertos, ”On integral representations of linear forms,”Rend. Circ. Mat. Palermo,46, 309–316 (1997). · Zbl 0893.46045 · doi:10.1007/BF02977031
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