scholarly journals A COMMENT ON JONES INCLUSIONS WITH INFINITE INDEX

1995 ◽  
Vol 07 (04) ◽  
pp. 599-630 ◽  
Author(s):  
FLORIAN NILL ◽  
HANS-WERNER WIESBROCK
Keyword(s):  
Fixed Point ◽  
General Result ◽  
Conditional Expectation ◽  
Von Neumann Algebras ◽  
Alternative Proof ◽  
Infinite Index ◽  
Von Neumann ◽  
Outer Action ◽  
Fixed Point Algebra

Given an irreducible inclusion of infinite von-Neumann-algebras [Formula: see text] together with a conditional expectation [Formula: see text] such that the inclusion has depth 2, we show quite explicitly how [Formula: see text] can be viewed as the fixed-point algebra of [Formula: see text] w.r.t. an outer action of a compact Kac algebra acting on [Formula: see text]. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest [6], see also S. Yamagami [28].

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Factoriality and Type Classification of k-Graph von Neumann Algebras

2016 ◽  
Vol 60 (2) ◽  
pp. 499-518 ◽  
Author(s):  
Dilian Yang
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Single Vertex ◽  
Tracial State ◽  
Graph Von Neumann Algebras ◽  
Fixed Point Algebra ◽  
Type Classification

AbstractLet be a single vertex k-graph and let be the von Neumann algebra induced from the Gelfand–Naimark–Segal (GNS) representation of a distinguished state ω of its k-graph C*-algebra . In this paper we prove the factoriality of , and furthermore determine its type when either has the little pullback property, or the intrinsic group of has rank 0. The key step to achieving this is to show that the fixed-point algebra of the modular action corresponding to ω has a unique tracial state.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

scholarly journals Infinite index extensions of local nets and defects

2018 ◽  
Vol 30 (02) ◽  
pp. 1850002 ◽  
Author(s):  
Simone Del Vecchio ◽  
Luca Giorgetti
Keyword(s):  
Gauge Theories ◽  
Von Neumann Algebras ◽  
Index Case ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Infinite Index ◽  
Von Neumann ◽  
Jones Index ◽  
The Family ◽  
Internal Symmetries

The subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [62] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi–Longo–Popa [44] and Fidaleo–Isola [30], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [58] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [7].

scholarly journals On pathological properties of fixed point algebras in Kirchberg algebras

2019 ◽  
Vol 150 (6) ◽  
pp. 3087-3096
Author(s):  
Yuhei Suzuki
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Approximation Property ◽  
Infinite Group ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Outer Action ◽  
Reduced Group ◽  
Fixed Point Algebra

AbstractWe investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

GEOMETRY OF CONDITIONAL EXPECTATIONS AND FINITE INDEX

1994 ◽  
Vol 05 (02) ◽  
pp. 169-178 ◽  
Author(s):  
ESTEBAN ANDRUCHOW ◽  
DEMETRIO STOJANOFF
Keyword(s):  
Homogeneous Space ◽  
Finite Index ◽  
Conditional Expectation ◽  
Von Neumann Algebras ◽  
Covering Space ◽  
Von Neumann ◽  
Similarity Orbit ◽  
Conditional Expectations

Let e be the Jones projection associated to a conditional expectation [Formula: see text] where [Formula: see text] are von Neumann algebras. We prove that the similarity orbit of e by invertibles of [Formula: see text] is an homogeneous space iff the index of E is finite. If also [Formula: see text], then this orbit is a covering space for the orbit of E.

Von Neumann algebras arising as fixed point algebras under xerox type actions

2014 ◽  
Vol 72 (2) ◽  
pp. 343-369 ◽  
Author(s):  
Thierry Giordano ◽  
Radu-B. Munteanu
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Von Neumann

scholarly journals On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

2019 ◽  
Vol 125 (2) ◽  
pp. 210-226
Author(s):  
Selçuk Barlak ◽  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Outer Action ◽  
Relative Commutant ◽  
Associated Matrix ◽  
Fixed Point Algebra

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.

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