Maximal extensions of a linear functional

Year 2023, Volume: 6 Issue: 4, 198 - 209, 15.12.2023

Fabio Burderi Camillo Trapanı Salvatore Triolo

https://doi.org/10.33205/cma.1310238

Abstract

Extensions of a positive hermitian linear functional $\omega$, defined on a dense *-subalgebra $\mathfrak{A_{0}}$ of a topological *-algebra $\mathfrak{A}[\tau]$ are analyzed. It turns out that their maximal extension as linear functionals or hermitian linear functional are everywhere defined. The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [1] is rivisited from this point of view. Examples mostly taken from the theory of integration are discussed.

Keywords

Positive linear functionals, Topological *-algebras, ;Extension of linear functionals

References

• A. Bikchentaev: The algebra of thin measurable operators is directly finite, Constr. Math. Anal., 6 (1) (2023), 1–5.
• F. Burderi, C. Trapani and S.Triolo: Extensions of hermitian linear functionals, Banach J. Math. Anal., 16 (3) (2022), 45.
• B. Bongiorno, C. Trapani and S.Triolo: Extensions of positive linea functionals on a Topological *-algebra, Rocky Mountain Journal of Mathematics, 40 (6) (2010), 1745–1777.
• O. Bratteli, D. W. Robinson: Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag, Berlin (1979).
• R. V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras, I, Academic Press, New York (1983).
• G. Köthe: Topological Vector Spaces, II, Springer-Verlag, New York (1979).
• J. Foran: An extension of the Denjoy integral, Proc. Amer. Math. Soc., 49 (1975), 359–365.
• R. A. Gordon: The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, American Mathematical Society, Providence (1994).
• J. Kurzweil: Nichtalsolut Konvergente Integrale, Teubner, Leipzig (1980).
• J. Lu, P.Y. Lee: The primitives of Henstock integrable functions in Euclidean space, Bull. London Math. Soc., 31 (1999), 173–180.
• E. Nelson: Note on non-commutative integration, J. Funct. Anal., 15 (1974), 103–116.
• C. La Russa, S. Triolo: Radon-Nikodym theorem in quasi *-algebras, J. Operator Theory, 69 (2) (2013), 423-–433.
• E. Malkowsky, V. Rakoˇcevi´c: Advanced Functional Analysis 1st Edition, (2019) ISBN 978-1138337152
• T. Ogasawara, K. Yoshinaga: A non commutative theory of integration for operators, J. Sci. Hiroshima Univ., 18 (1955), 312–347.
• W. Rudin: Real and Complex Analysis, Mc-Graw-Hill (1966).
• I. E. Segal: A noncommutative extension of abstract integration, Ann. Math., 57 (1953), 401–457.
• S. Triolo: WQ*-Algebras of measurable operators, Indian J. Pure Appl. Math., 43 (6) (2012), 601–617.
• S. Triolo: Possible extensions of the noncommutative integral, Rend. Circ. Mat. Palermo, 60 (3) (2011), 409–416.
• C. Trapani, S.Triolo: Representations of certain banach C∗− modules, Mediterr. J. Math., 1 (4) (2004), 441–461.
• G. Bellomonte, C. Trapani and S. Triolo: Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals., Mediterr. J. Math., 7 (2010), 63–74.