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}.

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].

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

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).

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 Outer actions of a discrete amenable group on approximately finite dimensional factors II, the III$_{\lambda}$-case, $\lambda\neq 0$

2007 ◽  
Vol 100 (1) ◽  
pp. 75 ◽  
Author(s):  
Yoshikazu Katayama ◽  
Masamichi Takesaki
Keyword(s):  
Abelian Group ◽  
Cohomology Group ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Classification Result ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Outer Action ◽  
The One ◽  
Abelian Von Neumann Algebra

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0<\lambda<1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

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.

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