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Maximal extensions of a linear functional

Yıl 2023, , 198 - 209, 15.12.2023
https://doi.org/10.33205/cma.1310238

Öz

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.

Kaynakça

  • 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.
Yıl 2023, , 198 - 209, 15.12.2023
https://doi.org/10.33205/cma.1310238

Öz

Kaynakça

  • 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.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Operatör Cebirleri ve Fonksiyonel Analiz
Bölüm Makaleler
Yazarlar

Fabio Burderi 0000-0002-1380-867X

Camillo Trapanı 0000-0001-9386-4403

Salvatore Triolo 0000-0002-9729-2657

Erken Görünüm Tarihi 28 Eylül 2023
Yayımlanma Tarihi 15 Aralık 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Burderi, F., Trapanı, C., & Triolo, S. (2023). Maximal extensions of a linear functional. Constructive Mathematical Analysis, 6(4), 198-209. https://doi.org/10.33205/cma.1310238
AMA Burderi F, Trapanı C, Triolo S. Maximal extensions of a linear functional. CMA. Aralık 2023;6(4):198-209. doi:10.33205/cma.1310238
Chicago Burderi, Fabio, Camillo Trapanı, ve Salvatore Triolo. “Maximal Extensions of a Linear Functional”. Constructive Mathematical Analysis 6, sy. 4 (Aralık 2023): 198-209. https://doi.org/10.33205/cma.1310238.
EndNote Burderi F, Trapanı C, Triolo S (01 Aralık 2023) Maximal extensions of a linear functional. Constructive Mathematical Analysis 6 4 198–209.
IEEE F. Burderi, C. Trapanı, ve S. Triolo, “Maximal extensions of a linear functional”, CMA, c. 6, sy. 4, ss. 198–209, 2023, doi: 10.33205/cma.1310238.
ISNAD Burderi, Fabio vd. “Maximal Extensions of a Linear Functional”. Constructive Mathematical Analysis 6/4 (Aralık 2023), 198-209. https://doi.org/10.33205/cma.1310238.
JAMA Burderi F, Trapanı C, Triolo S. Maximal extensions of a linear functional. CMA. 2023;6:198–209.
MLA Burderi, Fabio vd. “Maximal Extensions of a Linear Functional”. Constructive Mathematical Analysis, c. 6, sy. 4, 2023, ss. 198-09, doi:10.33205/cma.1310238.
Vancouver Burderi F, Trapanı C, Triolo S. Maximal extensions of a linear functional. CMA. 2023;6(4):198-209.