Abstract
The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \({\mathfrak{A}_0}\) of a topological *-algebra \({\mathfrak{A}}\), satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.
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J-P. Antoine, A. Inoue and C. Trapani , Partial *-Algebras and Their Operator Realizations. Mathematics and its Applications, vol. 553, Kluwer Academic Publishers, Dordrecht, 2002.
B. Bongiorno, C. Trapani and S. Triolo, Extensions of positive linear functionals on a topological *-algebra. Preprint, Palermo 2007 (Rocky Mountain J. Math., to appear).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I. Texts and Monographs in Physics. Springer-Verlag, New York- Heidelberg, 1979.
Nelson E.: Note on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)
K. Schmüdgen, Unbounded Operator Algebras and Representation Theory. Operator Theory: Advances and Applications, 37. Birkhuser Verlag, Basel, 1990.
Segal I.E.: A noncommutative extension of abstract integration. Ann. Math. 57, 401–457 (1953)
S. Strǎtilǎ and L. Zsidó, Lectures on von Neumann Algebras. Editura Academiei, Bucharest and Abacus Press, Tunbridge Wells, Kent, 1979.
Takesaki M.: Theory of Operator Algebras. I. Springer-Verlag, New York-Heidelberg (1979)
Trapani C.: *-Representations, seminorms and structure properties of normed quasi *-algebras. Studia Math. 186(1), 47–75 (2008)
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Bellomonte, G., Trapani, C. & Triolo, S. Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals. Mediterr. J. Math. 7, 63–74 (2010). https://doi.org/10.1007/s00009-010-0027-2
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DOI: https://doi.org/10.1007/s00009-010-0027-2