On C*-Diagonals

1986 ◽  
Vol 38 (4) ◽  
pp. 969-1008 ◽  
Author(s):  
Alexander Kumjian
Keyword(s):  
Equivalence Relation ◽  
Cohomology Class ◽  
Cartan Subalgebra ◽  
Von Neumann Algebras ◽  
Topological Invariant ◽  
Full Generality ◽  
Von Neumann ◽  
Abelian Subalgebras ◽  
Analogous Question ◽  
Definition Of

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.

scholarly journals Cartan subalgebras of regular extensions of von Neumann algebras

1988 ◽  
Vol 45 (2) ◽  
pp. 249-274
Author(s):  
Colin E. Sutherland
Keyword(s):  
Cartan Subalgebra ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Projective Representation ◽  
Von Neumann ◽  
Left Regular ◽  
Cartan Subalgebras ◽  
Inner Automorphisms ◽  
The Given

AbstractWe analyse the structure of a regular extension ℳ ⋊ γ, υQ of a von Neumann algebra ℳ by an action (modulo inner automorphisms) γ of a discrete group Q, and a nonabelian 2-cycle υ for γ, under the assumption that the “action” γ of Q is cocycle conjugate to an “action”, α which leaves globally invariant a cartan subalgebra of ℳ. we show that ℳ ⋊ γ, υQ is isomorphic with the algebra of the left regular projective representation of a certain discrete, non-principal groupoid ℜ V Q determined by the action of Q on the given cartan subalgebrs, where ℜ is the Takesaki relation associated to the pair (ℳ, ) we apply this description to give a decomposition of the regular representation of a group G into irreducibles, where G is a split extension of a type I group K by an abelian group Q, and work out the details of the author's earlier abstract plancherel theorem in the case when K is abelian.

scholarly journals KK-Theory and Spectral Flow in von Neumann Algebras

2012 ◽  
Vol 10 (2) ◽  
pp. 241-277 ◽  
Author(s):  
J. Kaad ◽  
R. Nest ◽  
A. Rennie
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Closed Ideal ◽  
Spectral Triple ◽  
Spectral Flow ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Definition Of

AbstractWe present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

scholarly journals Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

2004 ◽  
Vol 95 (1) ◽  
pp. 124 ◽  
Author(s):  
Volker Runde
Keyword(s):  
Abelian Subgroup ◽  
Von Neumann Algebras ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Connes Amenability ◽  
Definition Of ◽  
Dual Banach Algebra

Let $\mathcal A$ be a dual Banach algebra with predual $\mathcal A_*$ and consider the following assertions: (A) $\mathcal A$ is Connes-amenable; (B) $\mathcal A$ has a normal, virtual diagonal; (C) $\mathcal A_*$ is an injective $\mathcal A$-bimodule. For general $\mathcal A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $\mathcal A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $\mathcal A = B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.

scholarly journals Amalgamated free product type III factors with at most one Cartan subalgebra

2013 ◽  
Vol 150 (1) ◽  
pp. 143-174 ◽  
Author(s):  
Rémi Boutonnet ◽  
Cyril Houdayer ◽  
Sven Raum
Keyword(s):  
Free Product ◽  
Measure Space ◽  
Cartan Subalgebra ◽  
Von Neumann Algebras ◽  
Finite Subgroup ◽  
Standard Measure ◽  
Von Neumann ◽  
Cartan Subalgebras ◽  
Amalgamated Free Product ◽  
Normal States

AbstractWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

scholarly journals The analogue of Choi matrices for a class of linear maps on Von Neumann algebras

2015 ◽  
Vol 26 (02) ◽  
pp. 1550018 ◽  
Author(s):  
Erling Størmer
Keyword(s):  
Von Neumann Algebras ◽  
General Context ◽  
Von Neumann ◽  
Linear Maps ◽  
Definition Of

The definition of Choi matrices for linear maps on the n × n matrices is extended to factors, and the basic theorems for Choi matrices are proved in this more general context.

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