Some problems and results on reflexive algebras

1985 ◽  
pp. 324-330 ◽  
Author(s):  
E. Christopher Lance
Keyword(s):  
Reflexive Algebras

Characterization of derivations on reflexive algebras

2013 ◽  
Vol 61 (10) ◽  
pp. 1408-1418
Author(s):  
Runling An ◽  
Jinchuan Hou
Keyword(s):  
Reflexive Algebras

Jordan Modules and Jordan Ideals of Reflexive Algebras

2012 ◽  
Vol 74 (1) ◽  
pp. 123-136
Author(s):  
Pengtong Li ◽  
Fengjie Li
Keyword(s):  
Reflexive Algebras ◽  
Jordan Modules

scholarly journals NORMALIZERS OF OPERATOR ALGEBRAS AND REFLEXIVITY

2003 ◽  
Vol 86 (2) ◽  
pp. 463-484 ◽  
Author(s):  
A. KATAVOLOS ◽  
I. G. TODOROV
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Spectral Synthesis ◽  
Sufficient Conditions ◽  
Linear Structure ◽  
Linear Spaces ◽  
Von Neumann ◽  
Primary 47L35 ◽  
Borel Functions ◽  
Reflexive Algebras

The set of normalizers between von Neumann (or, more generally, reflexive) algebras $\mathcal{A}$ and $\mathcal{B}$ (that is, the set of all operators $T$ such that $T \mathcal{A} T^{\ast} \subseteq \mathcal{B}$ and $T^{\ast} \mathcal{B} T \subseteq \mathcal{A}$) possesses ‘local linear structure’: it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces $\mathcal{U}$ of operators satisfying $\mathcal{UU}^{\ast} \mathcal{U} \subseteq \mathcal{U}$ (also known as ternary rings of operators). Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form $\mathcal{U} = \{T : TL = \phi (L) T$ for all $L \in \mathcal{L}\}$ where $\mathcal{L}$ is a set of projections and $\phi$ a certain map defined on $\mathcal{L}$. A normalizing space consists of normalizers between appropriate von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian self-adjoint algebras consist of operators ‘supported’ on sets of the form $[f = g]$ where $f$ and $g$ are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.2000 Mathematical Subject Classification: 47L05 (primary), 47L35, 46L10 (secondary).

scholarly journals Universally reflexive algebras

1991 ◽  
Vol 157 ◽  
pp. 195-201 ◽  
Author(s):  
K.R. Fuller ◽  
W.K. Nicholson ◽  
J.F. Watters
Keyword(s):  
Reflexive Algebras

scholarly journals Generators of reflexive algebras

1975 ◽  
Vol 20 (2) ◽  
pp. 159-164
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Finitely Generated ◽  
Separable Space ◽  
Von Neumann ◽  
Underlying Space ◽  
Weakly Closed ◽  
Reflexive Algebras

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

Reflexive Bimodules

1989 ◽  
Vol 41 (4) ◽  
pp. 592-611 ◽  
Author(s):  
K. R. Fuller ◽  
W. K. Nicholson ◽  
J. F. Watters
Keyword(s):  
Vector Space ◽  
Finite Dimensional Algebra ◽  
Dimensional Vector ◽  
Dimensional Algebra ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Reflexive Algebra ◽  
Reflexive Algebras

If VK is a finite dimensional vector space over a field K and L is a lattice of subspaces of V, then, following Halmos [11], alg L is defined to be (the K-algebra of) all K-endomorphisms of V which leave every subspace in L invariant. If R ⊆ end(VK) is any subalgebra we define lat R to be (the sublattice of) all subspaces of VK which are invariant under every transformation in R. Then R ⊆ alg [lat R] and R is called a reflexive algebra when this is equality. Every finite dimensional algebra is isomorphic to a reflexive one ([4]) and these reflexive algebras have been studied by Azoff [1], Barker and Conklin [3] and Habibi and Gustafson [9] among others.

Lie and Jordan Ideals in Reflexive Algebras

2007 ◽  
Vol 59 (2) ◽  
pp. 189-206 ◽  
Author(s):  
Fangyan Lu ◽  
Xiuping Yu
Keyword(s):  
Reflexive Algebras
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