Algebraic Inner Derivations on Operator Algebras

1983 ◽  
Vol 35 (4) ◽  
pp. 710-723
Author(s):  
C. Robert Miers ◽  
John Phillips
Keyword(s):  
Positive Integer ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Simple Ring

Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A″). In particular, the strict result holds for any von Neumann algebra, A.

scholarly journals Coordinates for analytic operator algebras

1989 ◽  
Vol 31 (1) ◽  
pp. 31-47
Author(s):  
Baruch Solel
Keyword(s):  
Abelian Group ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Dual Group ◽  
Positive Semigroup ◽  
Analytic Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Analytic Algebra ◽  
Image Position

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].

On groups of automorphisms of operator algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 237-243 ◽  
Author(s):  
J. Moffat
Keyword(s):  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Operator Algebras ◽  
Amenable Group ◽  
Locally Compact Abelian Group ◽  
Von Neumann ◽  
Weakly Continuous ◽  
Continuous Unitary Representation ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).

scholarly journals Operator algebras related to Thompson's group F

2005 ◽  
Vol 79 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Paul Jolissaint
Keyword(s):  
Cyclic Group ◽  
Commutator Subgroup ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Natural Action ◽  
Von Neumann ◽  
Ratio Set ◽  
Thompson’S Group ◽  
Central Sequences ◽  
Abelian Von Neumann Algebra

AbstractLet F′ be the commutator subgroup of F and let Γ0 be the cyclic group generated by the first generator of F. We continue the study of the central sequences of the factor L(F′), and we prove that the abelian von Neumann algebra L(Γ0) is a strongly singular MASA in L(F). We also prove that the natural action of F on [0, 1] is ergodic and that its ratio set is {0} ∪ {2k; k ∞ Z}.

C*-ALGEBRAIC METHODS IN STATISTICAL MECHANICS AND FIELD THEORY

1990 ◽  
Vol 04 (05) ◽  
pp. 1069-1118 ◽  
Author(s):  
David E. EVANS
Keyword(s):  
Field Theory ◽  
Statistical Mechanics ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Inductive Limits ◽  
Von Neumann ◽  
C Algebras ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Af Algebras

We survey the recent work in non-commutative operator algebras (especially AF-algebras, those which are inductive limits of finite dimensional C*-algebras) and which arise in studying critical phenomena in classical statistical mechanics and conformal field theory, from a C*- or topological viewpoint, rather than a von Neumann algebra/measure theoretic one.

scholarly journals A von Neumann algebraic approach to self-similar group actions

2019 ◽  
Vol 30 (14) ◽  
pp. 1950074
Author(s):  
Keisuke Yoshida
Keyword(s):  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Group Actions ◽  
Algebraic Approach ◽  
Similar Group ◽  
Von Neumann ◽  
Kms State ◽  
Bernoulli Measure ◽  
Self Similar ◽  
Generic Points

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

scholarly journals Derivations on commutative operator algebras

1985 ◽  
Vol 32 (3) ◽  
pp. 415-418
Author(s):  
Mark Spivack
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Bounded Operator ◽  
Alternative Proof ◽  
Reflexive Banach Spaces ◽  
Von Neumann ◽  
Simple Alternative

It is well-known that any derivation on a commutative von Neumann algebra is implemented by a bounded operator. In this note we present a simple alternative proof, which generalizes the result further within Hilbert space, and to reflexive Banach spaces.

scholarly journals Causality in Schwinger’s Picture of Quantum Mechanics

Entropy ◽  
2022 ◽  
Vol 24 (1) ◽  
pp. 75
Author(s):  
Florio M. Ciaglia ◽  
Fabio Di Di Cosmo ◽  
Alberto Ibort ◽  
Giuseppe Marmo ◽  
Luca Schiavone ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Quantum Mechanical ◽  
Von Neumann ◽  
Triangular Operator ◽  
Quantum Mechanical Systems ◽  
Causal Structures ◽  
Incidence Theorem

This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

scholarly journals Jordan (α,β)-Derivations on Operator Algebras

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Quanyuan Chen ◽  
Xiaochun Fang ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Operator Algebras ◽  
Von Neumann

Let A be a CSL subalgebra of a von Neumann algebra acting on a Hilbert space H. It is shown that any Jordan (α,β)-derivation on A is an (α,β)-derivation, where α,β are any automorphisms on A. Moreover, the nth power (α,β)-maps on A are investigated.

AFD MULTIPLIER ALGEBRAS

2006 ◽  
Vol 17 (09) ◽  
pp. 1091-1102
Author(s):  
P. W. NG
Keyword(s):  
Von Neumann Algebra ◽  
Real Rank Zero ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Rank Zero ◽  
C Algebra ◽  
Unique Trace ◽  
Type Property ◽  
Unital Simple ◽  
Multiplier Algebras

We show that the multiplier algebra of a simple stable nuclear C*-algebra has a property similar to that of an AFD or hyperfinite von Neumann algebra. Specifically, we prove the following: Theorem 0.1. Let [Formula: see text] be a unital simple separable C*-algebra. Let [Formula: see text] be the multiplier algebra of the stabilization of [Formula: see text]. Then [Formula: see text] is nuclear if and only if [Formula: see text] has the AFD-type property. We also study a stronger property called the "strong AFD-type property". We show that if [Formula: see text] is a unital simple real rank zero AT-algebra with unique trace, then the multiplier algebra [Formula: see text] of the stabilization of [Formula: see text] has the strong AFD-type property, and we raise the question of whether this is true more generally.

scholarly journals Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

2011 ◽  
Vol 63 (4) ◽  
pp. 798-825 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Von Neumann Algebra ◽  
Space Structure ◽  
Fourier Algebra ◽  
Careful Study ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Isometric Representation ◽  
Operator Space Structure ◽  
Definition Of ◽  
The Right

Abstract We show that the multiplier algebra of the Fourier algebra on a locally compact group G can be isometrically represented on a direct sum on non-commutative Lp spaces associated with the right von Neumann algebra of G. The resulting image is the idealiser of the image of the Fourier algebra. If these spaces are given their canonical operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the noncommutative Lp spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca–Herz algebra built out of these non-commutative Lp spaces, say . It is shown that is isometric to L1(G), generalising the abelian situation.

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