On derivations of operator algebras implemented by bounded operators

1990 ◽  
pp. 163-174
Author(s):  
Edward V. Kissin
Keyword(s):  
Operator Algebras ◽  
Bounded Operators

An Inequality Characterizes the Trace

1979 ◽  
Vol 31 (6) ◽  
pp. 1322-1328 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Triangle Inequality ◽  
Operator Algebras ◽  
Schwarz Inequality ◽  
Unit Element ◽  
Integration Theory ◽  
List Type ◽  
Bounded Operators ◽  
Linear Maps ◽  
Image Position ◽  
Equivalent Conditions

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |ϕ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional ϕ on the C*-algebra satisfies(†)then ϕ satisfies also the equivalent conditions (i) ϕ(xy) = ϕ(yx) for all x, y in ;(ii) ϕ(x*x) = ϕ(xx*) for all x in ;(iii) ϕ(x) = ϕ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element.

scholarly journals Lie and Jordan structure in operator algebras

1980 ◽  
Vol 29 (2) ◽  
pp. 129-142
Author(s):  
Derek W. Robinson ◽  
Erling Størmer
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Operator Algebras ◽  
Bounded Operators ◽  
Jordan Structure ◽  
Spectral Subspaces

AbstractLet υ be a C*-algebra, α a *-anti-automorphism of order 2, and υα(±1) = {A; A ∈ υ, α(A) = ± A} the spectral subspaces of α. It follows that υα(+ 1) is a Jordan algebra and υα(− 1) is a Lie algebra. We begin the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining υ = ℒ(ℋ), the algebra of bounded operators on a Hilbert space ℋ.

scholarly journals Joint numerical ranges for unbounded normal operators

1985 ◽  
Vol 28 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Huang Danrun
Keyword(s):  
Unbounded Operator ◽  
Operator Algebras ◽  
Numerical Range ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Joint Spectrum ◽  
Normal Operators ◽  
Intimate Relations ◽  
Joint Numerical Range

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

Completely Reducible Operator Algebras and Spectral Synthesis

1982 ◽  
Vol 34 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Shlomo Rosenoer
Keyword(s):  
Operator Algebras ◽  
Spectral Synthesis ◽  
Invariant Subspaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Weakly Closed ◽  
Reductive Algebra

An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK* commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows.

scholarly journals Boundedness and completeness in locally convex spaces and algerbras

1977 ◽  
Vol 24 (1) ◽  
pp. 50-63 ◽  
Author(s):  
Gerard A. Joseph
Keyword(s):  
Convex Space ◽  
Operator Algebras ◽  
Bounded Operator ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Locally Convex ◽  
Convex Spaces

AbstractThe set of bounded elements of a unital l.m.c. algebra is characterised as the union of certain naturally defined normed subalgebras, and an analogous characterisation is given for algebras of quotient-bounded operators on a locally convex space. Pseudocomplete l.m.c. algebras are characterised in terms of the completeness of these subalgebras, and the equivalence of this condition with the pseudocompleteness of the quotient-bounded operator algebras established. The scalar multiples of the identity in a unital l.m.c. algebra are characterised in terms of certain boundedness conditions.

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.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

Z2k-code vertex operator algebras

2021 ◽  
Vol 573 ◽  
pp. 451-475
Author(s):  
Hiromichi Yamada ◽  
Hiroshi Yamauchi
Keyword(s):  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Operator Algebras
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