Boolean algebras of projections and uniformly closed operator algebras

1999 ◽  
pp. 41-56
Author(s):  
Werner Ricker
Keyword(s):  
Operator Algebras ◽  
Closed Operator ◽  
Boolean Algebras ◽  
Uniformly Closed

On Operator Algebras Containing Cyclic Boolean Algebras II

1977 ◽  
Vol s2-16 (3) ◽  
pp. 501-506 ◽  
Author(s):  
Peter Rosenthal ◽  
A. R. Sourour
Keyword(s):  
Operator Algebras ◽  
Boolean Algebras

Uniformly Closed Algebras Generated by Boolean Algebras of Projections in Locally Convex Spaces

1987 ◽  
Vol 39 (5) ◽  
pp. 1123-1146 ◽  
Author(s):  
Werner Ricker
Keyword(s):  
Operator Algebras ◽  
A Priori ◽  
Boolean Algebras ◽  
Locally Convex Spaces ◽  
Closed Algebra ◽  
Underlying Space ◽  
Weak Operator ◽  
Locally Convex ◽  
Convex Spaces

The theory of operator algebras in Banach spaces generated by Boolean algebras of projections is by now well known. It is systematically exposed in the penetrating studies of W. Bade, [1], [2] and [6, Chapter XVII]. Many of these results, a priori independent on normability of the underlying space, have recently been extended to the setting of locally convex spaces; see [3], [4], [5], [11] and [15], for example.However, one of Bade's fundamental results, stating that the closed algebra generated by a complete Boolean algebra in the uniform operator topology is the same as the closed algebra that it generates in the weak operator topology, has remained remarkably resistant in attempts to extend it to locally convex spaces. Recently however, a class of Boolean algebras in non-normable spaces, called boundedly σ-complete Boolean algebras, was exhibited in which the analogue of Bade's result is valid, [14; Theorem 5.3].

scholarly journals On operator algebras containing cyclic Boolean algebras

1977 ◽  
Vol 70 (1) ◽  
pp. 243-252 ◽  
Author(s):  
Peter Rosenthal ◽  
Ahmed Sourour
Keyword(s):  
Operator Algebras ◽  
Boolean Algebras

Closed Lie Ideals in Operator Algebras

1981 ◽  
Vol 33 (5) ◽  
pp. 1271-1278 ◽  
Author(s):  
C. Robert Miers
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Subspace ◽  
Operator Algebras ◽  
Lie Ideal ◽  
Von Neumann ◽  
Adjoint Operation ◽  
Uniformly Closed

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.

scholarly journals W algebras, cosets and VOAs for 4d $$ \mathcal{N} $$ = 2 SCFTs from M5 branes

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Dan Xie ◽  
Wenbin Yan
Keyword(s):  
Vertex Operator ◽  
Operator Algebras ◽  
Nilpotent Orbit ◽  
Vertex Operator Algebras ◽  
Flavor Symmetries ◽  
Conformal Embedding

Abstract We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the W algebras defined using nilpotent orbit with partition [qm, 1s]. Gauging above AD matters, we can find VOAs for more general $$ \mathcal{N} $$ N = 2 SCFTs engineered from 6d (2, 0) theories. For example, the VOA for general (AN − 1, Ak − 1) theory is found as the coset of a collection of above W algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for known VOAs can be derived from 4d theory.

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