Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2023, Cilt: 6 Sayı: 1, 1 - 5, 15.03.2023
https://doi.org/10.33205/cma.1181495

Öz

Proje Numarası

075-02-2022-882

Kaynakça

  • S. K. Berberian: Baer ${}^*$rings. Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin (1972).
  • S. K. Berberian: The algebra of thin operators is directly finite, Publ. Sec. Mat. Univ. Autònoma Barcelona, 26 (2) (1982), 5-7.
  • A. M. Bikchentaev: Local convergence in measure on semifinite von Neumamn algebras, Proc. Steklov Inst. Math., 255 (2006), 35-48.
  • A. M. Bikchentaev: On normal $\tau$-measurable operators affiliated with semifinite von Neumann algebras, Math. Notes, 96 (3-4) (2014), 332-341.
  • A. M. Bikchentaev: Concerning the theory of $\tau$-measurable operators affiliated to a semifinite von Neumann algebra, Math. Notes, 98 (3-4) (2015), 382-391.
  • A. M. Bikchentaev: On idempotent $\tau$-measurable operators affiliated to a von Neumann algebra, Math. Notes, 100 (3-4) (2016), 515-525.
  • A. M. Bikchentaev: On $\tau$-essentially invertibility of $\tau$-measurable operators, Internat. J. Theoret. Phys., 60 (2) (2021), 567-575.
  • A. M. Bikchentaev: Essentially invertible measurable operators affiliated to a semifinite von Neumann algebra and commutators, Sib. Math. J., 63 (2) (2022), 224-232.
  • P. Dodds, B. de Pagter: Normed Köthe spaces: A non-commutative viewpoint, Indag. Math. (N.S.), 25 (2) (2014), 206-249.
  • T. Fack, H. Kosaki: Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math., 123 (2) (1986), 269-300.
  • K. R. Goodearl: von Neumann regular rings, Monographs and Studies in Mathematics, vol. 4. Pitman (Advanced Publishing Program), Boston, Mass.-London (1979).
  • I. Halperin: On a theorem of Sterling Berberian, C. R. Math. Rep. Acad. Sci. Canada, 3 (1) (1981), 33-35.
  • I. Kaplansky: Rings of operators, W.A. Benjamin, Inc., New York-Amsterdam (1968).
  • E. Nelson: Notes on non-commutative integration, J. Funct. Anal., 15 (2) (1974), 103-116.
  • K. Saitô: On the algebra of measurable operators for a general $AW^{\ast} $-algebra. II, Tohoku Math. J. (2), 23 (3) (1971), 525-534.
  • I. E. Segal: A non-commutative extension of abstract integration, Ann. Math., 57 (3) (1953), 401-457.
  • I. D. Tembo: Invertibility in the algebra of $\tau$-measurable operators, in: Operator algebras, operator theory and applications, Oper. Theory Adv. Appl., vol. 195, Birkhäuser Verlag, Basel (2010), 245-256.
  • O. E. Tikhonov: Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Soviet Math. (Iz. VUZ), 31 (1) (1987), 110-114.

The algebra of thin measurable operators is directly finite

Yıl 2023, Cilt: 6 Sayı: 1, 1 - 5, 15.03.2023
https://doi.org/10.33205/cma.1181495

Öz

Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\lambda I$
with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. It is proved that every operator of $T(\mathcal{M},\tau)$ that is left-invertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$.
It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$.
For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$, the inclusion $\mu(t; Q)\in \{0\}\bigcup
[1, +\infty)$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

Destekleyen Kurum

Volga Region Mathematical Center

Proje Numarası

075-02-2022-882

Kaynakça

  • S. K. Berberian: Baer ${}^*$rings. Die Grundlehren der mathematischen Wissenschaften, Band 195, Springer-Verlag, New York-Berlin (1972).
  • S. K. Berberian: The algebra of thin operators is directly finite, Publ. Sec. Mat. Univ. Autònoma Barcelona, 26 (2) (1982), 5-7.
  • A. M. Bikchentaev: Local convergence in measure on semifinite von Neumamn algebras, Proc. Steklov Inst. Math., 255 (2006), 35-48.
  • A. M. Bikchentaev: On normal $\tau$-measurable operators affiliated with semifinite von Neumann algebras, Math. Notes, 96 (3-4) (2014), 332-341.
  • A. M. Bikchentaev: Concerning the theory of $\tau$-measurable operators affiliated to a semifinite von Neumann algebra, Math. Notes, 98 (3-4) (2015), 382-391.
  • A. M. Bikchentaev: On idempotent $\tau$-measurable operators affiliated to a von Neumann algebra, Math. Notes, 100 (3-4) (2016), 515-525.
  • A. M. Bikchentaev: On $\tau$-essentially invertibility of $\tau$-measurable operators, Internat. J. Theoret. Phys., 60 (2) (2021), 567-575.
  • A. M. Bikchentaev: Essentially invertible measurable operators affiliated to a semifinite von Neumann algebra and commutators, Sib. Math. J., 63 (2) (2022), 224-232.
  • P. Dodds, B. de Pagter: Normed Köthe spaces: A non-commutative viewpoint, Indag. Math. (N.S.), 25 (2) (2014), 206-249.
  • T. Fack, H. Kosaki: Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math., 123 (2) (1986), 269-300.
  • K. R. Goodearl: von Neumann regular rings, Monographs and Studies in Mathematics, vol. 4. Pitman (Advanced Publishing Program), Boston, Mass.-London (1979).
  • I. Halperin: On a theorem of Sterling Berberian, C. R. Math. Rep. Acad. Sci. Canada, 3 (1) (1981), 33-35.
  • I. Kaplansky: Rings of operators, W.A. Benjamin, Inc., New York-Amsterdam (1968).
  • E. Nelson: Notes on non-commutative integration, J. Funct. Anal., 15 (2) (1974), 103-116.
  • K. Saitô: On the algebra of measurable operators for a general $AW^{\ast} $-algebra. II, Tohoku Math. J. (2), 23 (3) (1971), 525-534.
  • I. E. Segal: A non-commutative extension of abstract integration, Ann. Math., 57 (3) (1953), 401-457.
  • I. D. Tembo: Invertibility in the algebra of $\tau$-measurable operators, in: Operator algebras, operator theory and applications, Oper. Theory Adv. Appl., vol. 195, Birkhäuser Verlag, Basel (2010), 245-256.
  • O. E. Tikhonov: Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Soviet Math. (Iz. VUZ), 31 (1) (1987), 110-114.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Airat Bikchentaev 0000-0001-5992-3641

Proje Numarası 075-02-2022-882
Yayımlanma Tarihi 15 Mart 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 6 Sayı: 1

Kaynak Göster

APA Bikchentaev, A. (2023). The algebra of thin measurable operators is directly finite. Constructive Mathematical Analysis, 6(1), 1-5. https://doi.org/10.33205/cma.1181495
AMA Bikchentaev A. The algebra of thin measurable operators is directly finite. CMA. Mart 2023;6(1):1-5. doi:10.33205/cma.1181495
Chicago Bikchentaev, Airat. “The Algebra of Thin Measurable Operators Is Directly Finite”. Constructive Mathematical Analysis 6, sy. 1 (Mart 2023): 1-5. https://doi.org/10.33205/cma.1181495.
EndNote Bikchentaev A (01 Mart 2023) The algebra of thin measurable operators is directly finite. Constructive Mathematical Analysis 6 1 1–5.
IEEE A. Bikchentaev, “The algebra of thin measurable operators is directly finite”, CMA, c. 6, sy. 1, ss. 1–5, 2023, doi: 10.33205/cma.1181495.
ISNAD Bikchentaev, Airat. “The Algebra of Thin Measurable Operators Is Directly Finite”. Constructive Mathematical Analysis 6/1 (Mart 2023), 1-5. https://doi.org/10.33205/cma.1181495.
JAMA Bikchentaev A. The algebra of thin measurable operators is directly finite. CMA. 2023;6:1–5.
MLA Bikchentaev, Airat. “The Algebra of Thin Measurable Operators Is Directly Finite”. Constructive Mathematical Analysis, c. 6, sy. 1, 2023, ss. 1-5, doi:10.33205/cma.1181495.
Vancouver Bikchentaev A. The algebra of thin measurable operators is directly finite. CMA. 2023;6(1):1-5.