Operator valued weights in von Neumann algebras,II

Let M be a von Neumann algebra, and N a sub von Neumann algebra of M. We prove that if φ and ψ are n.f.s. weights on N and M respectively, such that σ_t^ψ extends σ_t^φ, then there is a unique n.f.s. operator valued weight T from M to N, such that ψ = φ ° T. Moreover we generalize the notion of modular automorphism groups associated with conditional expectations to operator valued weights.     

Invariant subspaces and cocycles in nonselfadjoint crossed products

Let G be a compact abelian group with the archimedean totally ordered dual Γ and let $\text{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a one-parameter group {α_γ}_γ?Γ of trace preserving 1-automorphisms of M. In this paper, we investigate the structure of invariant subspaces and cocycles for the subalgebra $\text{L}$₊ of $\text{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group {β_g}_g?G on $\text{L}$ is nonnegative. Our main result asserts that if M is a factor, then $\text{L}$₊ is maximal among the σ-weakly closed subalgebras of $\text{L}$.     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

Values of the Pukánszky invariant in McDuff factors

In 1960 Pukánszky introduced an invariant associating to every masa in a separable II₁ factor a non-empty subset of N∪{∞}. This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that any non-empty subset of N∪{∞} arises as the Pukánszky invariant of some masa in a separable McDuff II₁ factor containing a masa with Pukánszky invariant {1}. In particular the hyperfinite II₁ factor and all separable McDuff II₁ factors with a Cartan masa satisfy this hypothesis. In a general separable McDuff II₁ factor we show that every subset of N∪{∞} containing ∞ is obtained as a Pukánszky invariant of some masa.     

Representations of product systems over semigroups and dilations of commuting CP maps

We prove that every pair of commuting CP maps on a von Neumann algebra M can be dilated to a commuting pair of endomorphisms (on a larger von Neumann algebra). To achieve this, we first prove that every completely contractive representation of a product system of C^∗-correspondences over the semigroup N² can be dilated to an isometric (or Toeplitz) representation.     

Every compact group arises as the outer automorphism group of a II1 factor

We show that any compact group can be realized as the outer automorphism group of a factor of type II₁. This has been proved in the abelian case by Ioana, Peterson and Popa [A. Ioana, J. Peterson, S. Popa, Amalgamated free products of w-rigid factors and calculation of their symmetry group, math.OA/0505589, Acta Math., in press] applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs.     

A semi-finite algebra associated to a subfactor planar algebra

We canonically associate to any planar algebra two type II_∞ factors M_±. The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of M_± to finite projections. We show that each M_± is isomorphic to an amalgamated free product of type I von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that M₊≅M₋ is the amplification a free group factor on a finite number of generators. As an application, we show that the factors Mj constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(rj)), rj=1+2δ−2j(δ−1)I, where δ² is the index of the planar algebra and I is its global index. Other applications include computations of laws of Jones-Wenzl projections.     

Operator valued weights in von Neumann algebras,I

An operator valued weight is a kind of generalized conditional expectation from a von Neumann algebra M to a sub von Neumann algebra N. If T is a n.f.s. (normal, faithful, semifinite) operator valued weight from M to N, and φ is a n.f.s. weight on N, then φ ° T defines a n.f.s. weight on M. Our main result is that σ_t^φ°T extends σ_t^φ, and that the map φ → φ ° T preserves cocycle Radon Nikodym derivatives.     

Some quasinilpotent generators of the hyperfinite II1 factor

For each sequence n{cn} in l₁(N) we define an operator A in the hyperfinite II₁-factor R. We prove that these operators are quasinilpotent and they generate the whole hyperfinite II₁-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A^∗A.     

Almost periodic states and factors of type III1

We construct a factor of type III₁ which has no almost-periodic state (or weight). We exhibit a factor N of type II_∞ and two automorphisms θ₁θ₂ of N which are not in the same conjugacy class in Out $\text{N =}\text{Aut}\text{N}\text{Int}\text{N}$ though τθ₁ = λ_τ, τθ₂ = λ_τ with λ? ]0, 1[, τ = Trace on N. We introduce and study two invariants Sd and τ for factors of type III₁. We relate the closedness of Int M in Aut M to the absence of central sequences in the von Neumann algebra M.     

Maximality of entropy in finite von Neumann algebras

It is shown that the entropy function H(N ₁,…,N _k ) on finite dimensional von Neumann subalgebras of a finite von Neumann algebra attains its maximal possible value H(⋁ℓ=1k N _ℓ) if and only if there exists a maximal abelian subalgebra A of ⋁ℓ=1k N _ℓ such that A=⋁ℓ=1k(A∩N _ℓ). Oblatum 24-IV-1997 & 6-V-1997     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

From graphs to free products

We investigate a construction (from Kodiyalam Vijay and Sunder V?S, J.?Funct. Anal. 260 (2011) 2635?C2673) which associates a finite von Neumann algebra M(??,??) to a finite weighted graph (??,??). Pleasantly, but not surprisingly, the von Neumann algebra associated to a ??flower with n petals?? is the group on Neumann algebra of the free group on n generators. In general, the algebra M(??,??) is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs ??with one edge?? (or actually a pair of dual edges). This also yields ??natural?? examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) ${\mathbb C} \oplus {\mathbb C}$ -valued circular and semi-circular operators.     

The Unitary Implementation of a Locally Compact Quantum Group Action

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When α is an action of the locally compact quantum group (M, Δ) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N^αNM_αN is a basic construction. Here N^α denotes the fixed point algebra and M_αN is the crossed product. When α is an outer and integrable action on a factor N we prove that the inclusion N^αN is irreducible, of depth 2 and regular, giving a converse to the results of M. Enock and R. Nest (1996, J. Funct. Anal.137, 466–543; 1998, J. Funct. Anal.154, 67–109). Finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand.84, 297–319): every integrable outer action with infinite fixed point algebra is a dual action.     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

AlgebraicK-theory of von Neumann algebras

To every von Neumann algebra, one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the Abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II₁-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II₁-factors. As a corollary, it follows that the determinant induces an injection from the algebraicK ₁-group of the von Neumann algebra into the Abelian group of the invertible elements of the center. Its image is described. Another group,K ₁ ^w (A), which is generated by elements in matrix algebras overA that induce injective right multiplication maps, is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group Wh(G).Partially supported by NSF Grant DMS-9103327.     

Similarity degrees for the crossed product of von Neumann algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.