A reflexivity theorem for weakly closed subspaces of operators

Author:
Hari Bercovici

Journal:
Trans. Amer. Math. Soc. **288** (1985), 139-146

MSC:
Primary 47D15; Secondary 47A15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0773052-3

MathSciNet review:
773052

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It was proved in [**4**] that the ultraweakly closed algebras generated by certain contractions on Hilbert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to , was used in [**3**] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with , and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [**3**,**2** and **9**] as particular cases.

**[1]**C. Apostol, H. Bercovoci, C. Foias and C. Pearcy,*Invariant subspaces, dilation theory, and the structure of the predual of a dual operator algebra*. I, J. Functional Anal. (to appear).**[2]**Hari Bercovici, Bernard Chevreau, Ciprian Foias, and Carl Pearcy,*Dilation theory and systems of simultaneous equations in the predual of an operator algebra. II*, Math. Z.**187**(1984), no. 1, 97–103. MR**753424**, https://doi.org/10.1007/BF01163170**[3]**H. Bercovici, C. Foiaş, J. Langsam, and C. Pearcy,*(BCP)-operators are reflexive*, Michigan Math. J.**29**(1982), no. 3, 371–379. MR**674290****[4]**H. Bercovici, C. Foias, and C. Pearcy,*Factoring trace-class operator-valued functions with applications to the class 𝒜_{ℵ₀}*, J. Operator Theory**14**(1985), no. 2, 351–389. MR**808297****[5]**H. Bercovici, C. Foias, and C. Pearcy,*Dilation theory and systems of simultaneous equations in the predual of an operator algebra. I*, Michigan Math. J.**30**(1983), no. 3, 335–354. MR**725785**, https://doi.org/10.1307/mmj/1029002909**[6]**Arlen Brown and Carl Pearcy,*Introduction to operator theory. I*, Springer-Verlag, New York-Heidelberg, 1977. Elements of functional analysis; Graduate Texts in Mathematics, No. 55. MR**0511596****[7]**D. W. Hadwin and E. A. Nordgren,*Subalgebras of reflexive algebras*, J. Operator Theory**7**(1982), no. 1, 3–23. MR**650190****[8]**A. N. Loginov and V. S. Šul′man,*Hereditary and intermediate reflexivity of 𝑊*-algebras*, Izv. Akad. Nauk SSSR Ser. Mat.**39**(1975), no. 6, 1260–1273, 1437 (Russian). MR**0405124****[9]**Greg Robel,*On the structure of (BCP)-operators and related algebras. I*, J. Operator Theory**12**(1984), no. 1, 23–45. MR**757111**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47D15,
47A15

Retrieve articles in all journals with MSC: 47D15, 47A15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0773052-3

Article copyright:
© Copyright 1985
American Mathematical Society