scholarly journals Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations

1993 ◽  
Vol 107 (3) ◽  
pp. 273-286 ◽  
Author(s):  
Jan Rusinek
Keyword(s):  
Lie Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Lie Algebra Representations

Derived Ring Isomorphisms of Von Neumann Algebras

1973 ◽  
Vol 25 (6) ◽  
pp. 1254-1268 ◽  
Author(s):  
C. Robert Miers
Keyword(s):  
Scalar Field ◽  
Lie Algebra ◽  
Von Neumann Algebras ◽  
Linear Subspace ◽  
Complex Scalar ◽  
Von Neumann ◽  
Linear Map ◽  
Complex Scalar Field

Let M be an associative *-algebra with complex scalar field. M may be turned into a Lie algebra by defining multiplication by [A, B] = AB - BA. A Lie *-subalgebra L of M is a *-linear subspace of M such that if A, B ∈ L then [A,B] ∈ L. A Lie *-isomorphism ϕ between Lie *-subalgebras L1 and L2 of *-algebras M and N is a one-one, *-linear map of L1 onto L2 such that ϕ[A, B] = [ϕ(A), ϕ(B)] for all A , B ∈ L1.

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.

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann
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