A criterion for the spatial projectivity of an indecomposable CSL-algebra of operators

1994 ◽  
Vol 49 (4) ◽  
pp. 161-162 ◽  
Author(s):  
Yu O Golovin
Keyword(s):  
Csl Algebra ◽  
Algebra Of Operators

scholarly journals Note on amenable algebras of operators

1995 ◽  
Vol 52 (2) ◽  
pp. 327-329 ◽  
Author(s):  
P.C. Curtis ◽  
R.J. Loy
Keyword(s):  
C Algebra ◽  
Amenable Algebra ◽  
Algebra Of Operators

It is shown that an amenable algebra of operators on Hibert space which is generated by its normal elements is necessarily self-adjoint, so it is a C*-algebra.

scholarly journals Periodicity of functionals and representations of normed algebras on reflexive spaces

1976 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Hilbert Space ◽  
Continuous Homomorphism ◽  
Linear Operators ◽  
The Other ◽  
Continuous Linear ◽  
Normed Algebra ◽  
Reflexive Space ◽  
To Come ◽  
Algebra Of Operators ◽  
Continuous Linear Operators

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

The Tensor Product Formula for Reflexive Subspace Lattices

1995 ◽  
Vol 38 (3) ◽  
pp. 308-316 ◽  
Author(s):  
K. J. Harrison
Keyword(s):  
Tensor Product ◽  
Operator Algebra ◽  
Product Formula ◽  
Completely Distributive ◽  
Reflexive Subspace ◽  
Complemented Lattice ◽  
Csl Algebra ◽  
Subspace Lattices

AbstractWe give a characterisation of where and are subspace lattices with commutative and either completely distributive or complemented. We use it to show that Lat is a CSL algebra with a completely distributive or complemented lattice and is any operator algebra.

Perturbation of a nest algebra module

1983 ◽  
Vol 93 (2) ◽  
pp. 303-306 ◽  
Author(s):  
Sotirios Karanasios
Keyword(s):  
General Situation ◽  
Compact Perturbation ◽  
Nest Algebra ◽  
Nest Algebras ◽  
Weakly Closed ◽  
Carry Over ◽  
Algebra Module ◽  
Algebra Of Operators

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

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