This paper formulates Young-type inequalities for singular values (or $s$-numbers) and traces in the context of von Neumann algebras. In particular, it shown that if $\t(\cdot)$ is a faithful semifinite normal trace on a semifinite von Neumann algebra $M$ and if $p$ and $q$ are positive real numbers for which $p^{-1}+q^{-1}=1$, then, for all positive operators $a,b\in M$, $\t(|ab|)\le p^{-1}\t(a^p)+ q^{-1}\t(b^q)$, with equality holding (in the cases where $p^{-1}\t(a^p)+ q^{-1}\t(b^q)<\infty$) if and only if $b^q=a^p$.
In this paper we describe the amalgamated free product of finite and semifinite hyperfinite von Neumann algebras over atomic type I subalgebras. To do this we extend the notions of free dimension and standard embeddings used in the related results for finite von Neumann algebras to ones which work better for the semifinite case. We also define classes R3 (of finite von Neumann algebras) and R4 (of semifinite von Neumann algebras) which are closed under such amalgamated free products.
In this paper, we prove a non-commutative version of the Weyl-von Neumann theorem for representations of unital, separable AH algebras into countably decomposable, semifinite, properly infinite, von Neumann factors, where an AH algebra means an approximately homogeneous ${\rm C}^{\ast}$-algebra. We also prove a result for approximate summands of representations of unital, separable AH algebras into finite von Neumann factors.
In the current paper, we generalize the "compact operator" part of the Voiculescu's non-commutative Weyl-von Neumann theorem on approximate equivalence of unital $*$-homomorphisms of an commutative C$^*$ algebra $\mathcal{A}$ into a semifinite von Neumann algebra. A result of D. Hadwin for approximate summands of representations into a finite von Neumann factor $\mathcal{R}$ is also extended.
In this paper, we provide a generalized version of the Voiculescu theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful normal semifinite tracial weight $τ$, a normal operator is an arbitrarily small $(\max\{\|\cdot\|, \Vert\cdot\Vert_{2}\})$-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo norm ideals satisfying a natural condition.
We introduce a class of unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norms with repect to $τ$ on a von Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracial weight $τ$. We prove a Beurling-Chen-Hadwin-Shen theorem for $H^\infty$-invariant spaces of $L^α(\mathcal{M},τ)$, where $α$ is a unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norm with respect to $τ$, and $H^\infty$ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the $H^\infty$-invariant subspaces of a noncommutati…
In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $τ$ be a faithful normal semifinite tracial weight of $\mathcal M$. Suppose that $H$ and $H_1$ are self-adjoint operators affiliated with $\mathcal{M}$. We show that if $H-H_1$ is in $\mathcal{M}\cap L^{1}\left(\mathcal{M},τ\right)$, then the ${norm}$ absolutely continuous parts of $H$ and $H_1$ are unitarily equivalent. This implies that the real par…
In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We prove that a shift-monotone space $E(\Z)$ of sequences indexed by $\Z$ defines a Calkin space $E(\cM,τ)$ of $τ$-measurable operators affiliated with a semifinite von Neumann algebra $\cM$ equipped with a faithful normal semifinite trace $τ$. Furthermore, we show that shift-invariant functionals on $E(\Z)$ generate symmetric functionals on $E(\cM,τ)$. In the special case, when the algebra $\cM$ is atomless or atomic with atoms of equal trace, the converse also hold…
In 2008, Blecher and Labuschagne extended Beurling's classical theorem to $H^\infty$-invariant subspaces of $L^p(\mathcal{M},τ)$ for a finite von Neumann algebra $\mathcal{M}$ with a finite, faithful, normal tracial state $τ$ when $1\le p\le \infty$. In this paper, using Arveson's non-commutative Hardy space $H^\infty$ in relation to a von Neumann algebra $\mathcal{M}$ with a semifinite, faithful, normal tracial weight $τ$, we prove a Beurling-Blecher-Labuschagne theorem for $H^\infty$-invariant spaces of $L^p(\mathcal{M},τ)$ when $0<p\leq\infty$. The proof of the main result relies on proofs …
We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued $L_1$-space.
We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.
We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow of the path coincides with the value of trace on the associated K-class. Given a semifinite spectral triple (A,H,D) relative to a semifinite von Neumann algebra N, we construct a class [D] in KK^1(A,N') such that, for a unitary u in A, the von Neumann spectral flow between D and u*Du is equal to the Kasparo…
Utilizing the notion of uniform equicontinuity for sequences of functions with the values in the space of measurable operators, we present a non-commutative version of the Banach Principle for $L^\infty$.
In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an operator $a\in \mathcal M$ is said to be {irreducible in} $\mathcal M$ if $W^*(a)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(a)'\cap \mathcal M={\mathbb C} \cdot I$. In this paper, let $Φ(\cdot)$ be a $\Vert\cdot\Vert$-dominating, unitarily invariant norm (see Definition 2.1), where by $\Vert\cdot\Vert$ we denote the operator norm. We prove that in eve…
Recently Kosaki proved an uncertainty principle for matrices, related to Wigner-Yanase-Dyson information, and asked if a similar inequality could be proved in the von Neumann algebra setting. In this paper we prove such an uncertainty principle in the semifinite case.
We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is `almost' a (b,B)-cocycle in the cyclic cohomology of \A.
The theory of non symmetric Dirichlet forms is generalized to the non abelian setting, also establishing the natural correspondences among Dirichlet forms, sub-Markovian semigroups and sub-Markovian resolvents within this context. Examples of non symmetric Dirichlet forms given by derivations on Hilbert algebras are studied.
We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x)\to 0$ as $t\to\infty$ for every $x\in E$, where $μ_t(x)$ is the non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmet…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge bilaterally almost uniformly. We show that these averages converge almost uniformly in each noncommutative symmetric space $E$ such that $μ_t(x) \to 0$ as $t \to 0$ for every $x \in E$, where $μ_t(x)$ is a non-increasing rearrangement of $x$. Noncommutative Dunford-Schwartz-type multiparameter ergodic theorems are studied. A wide range of noncommutative symmetric …
We prove some structure results for isometries between noncommutative Lp spaces associated to von Neumann algebras. We find that an isometry T: Lp(M_1) to Lp(M_2) (1 le p < infty, p not 2) can be canonically expressed in a certain simple form whenever M_1 has variants of Watanabe's extension property. Conversely, this form always defines an isometry provided that M_1 is "approximately semifinite" (defined below). Although neither of these properties is fully understood, we show that they are enjoyed by all semifinite algebras and hyperfinite algebras (with no summand of type I_2), plus others.…
Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},τ)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $δ:\mathcal{A}\to E(\mathcal{M},τ)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The Følner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$.
Banach non-isomorphism of type III and semifinite noncommutative $L_p$-spaces
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学学院）；School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）
We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.
We introduce "embedding dimensions" of a family of generators of a finite von Neumann algebra when the von Neumann algebra can be faithfully embedded into the ultrapower of the hyperfinite II$_1$ factor. These embedding dimensions are von Neumann algebra invariants, i.e., do not depend on the choices of the generators. We also find values of these invariants for some specific von Neumann algebras.
In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let N and M be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner-Popa. If N and M are sufficiently close, then N and M are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.
We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Gamma and prime. The proof is based on C*-algebra theory.
The paper is devoted to the description of $2$-local derivations on von Neumann algebras. Earlier it was proved that every $2$-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type $III$ von Neumann algebras. This implies that on arbitrary von Neumann algebra each $2$-local derivation is a derivation.
Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or $W^*$-objects among the Hilbert ($C^*$-)modules are around since the first papers by Paschke (1973) and Rieffel (1974) that lift Kaplansky's setting (1953) to modules over noncommutative $C^*$-algebras. While the formal definition of $W^*$-modules} is due to Baillet, Denizeau, and Havet (1988), the one of von Neumann modules as strongly closed operator spaces started with Skeide (2000). It has been paired with the definition…
The Stone spectrum of a von Neumann algebra is a generalization of the Gelfand spectrum, as was shown by de Groote. In this article we clarify the structure of the Stone spectra of von Neumann algebras of type $I_{n}$.
In this paper we describe the amalgamated free product of two hyperfinite von Neumann algebras over a finite dimensional subalgebra. In general the free product is a finite direct sum of interpolated free group factors and a hyperfinite von Neumann algebra. We then show that the class of von Neumann algebras of this form is closed under taking amalgamated free products over finite dimensional subalgebras.
We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations.
In this paper, we discuss some equivalent definitions of Property $Γ$ for a type II$_1$ von Neumann algebra. Using these equivalent definitions, we prove that the Pisier's similarity degree of a type II$_1$ von Neumann algebra with Property $Γ$ is equal to $3$.
We apply the ideas of Muhly, Skeide and Solel [MSS03] of considering von Neumann B-modules as intertwiner spaces for representations of B' to obtain a new, simple and self-contained proof for self-duality of von Neumann modules. This simplifies also the approach of [MSS03].
Using Speicher central limit Theorem we provide Hiai's q-Araki-Woods von Neumann algebras with nice asymptotic matricial models. Then, we use this model and an elaborated ultraproduct procedure, to show that all q-Araki-Woods von Neumann algebras are QWEP.
We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$
It is shown that if a bipartite behavior admits a field representation in which Alice (or Bob's) observable algebra generates a purely atomic von Neumann algebra then it is non-relativistic.
In the sequel we establish the Banach Principle for semifinite JW-algebras without direct summand of type I sub 2, which extends the recent results of Chilin and Litvinov on the Banach Principle for semifinite von Neumann algebras to the case of JW-algebras.
A theory of Galois co-objects for von Neumann bialgebras is introduced. This concept is closely related to the notion of comonoidal W*-Morita equivalence between von Neumann bialgebras, which is a Morita equivalence taking the comultiplication structure into account. We show that the property of `being a von Neumann algebraic quantum group' (i.e. `having invariant weights') is preserved under this equivalence relation. We also introduce the notion of a projective corepresentation for a von Neumann bialgebra, and show how it leads to a construction method for Galois co-objects and comonoidal W*…
We prove that every surjective isometry between the unit spheres of two von Neumann algebras admits a unique extension to a surjective real linear isometry between these two algebras.
We define $Γ_q(B,S \otimes H)$, the generalized $q$-gaussian von Neumann algebras associated to a sequence of symmetric independent copies $(π_j,B,A,D)$ and to a subset $1 \in S = S^* \subset A$ and, under certain assumptions, prove their strong solidity relative to $B$. We provide many examples of strongly solid generalized $q$-gaussian von Neumann algebras. We also obtain non-isomorphism and non-embedability results about some of these von Neumann algebras.
We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.
We show that the second cohomology group $H^2(M\overline\otimes N, M\overline\otimes N)$ is always zero for arbitrary type II$_1$ von Neumann algebras $M$ and $N$.
