Gilles Pisier
This paper is mainly devoted to the following question:\ Let $M,N$ be von~Neumann algebras with $M\subset N$, if there is a completely bounded projection $P\colon \ N\to M$, is there automatically a contractive projection $\widetilde P\colon \ N\to M$? We give an affirmative answer with the only restriction that $M$ is assumed semi-finite.
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In this paper we calculate some geometric constants for Morrey spaces and small Morrey spaces, namely generalized Von Neumann-Jordan constant, modified Von Neumann-Jordan constants, and Zbáganu constant. All these constants measure the uniformly nonsquareness of the spaces. We obtain that their values are the same as the value of Von Neumann-Jordan constant for Morrey spaces and small Morrey spaces.
Upper and lower estimates are obtained for the Schatten-von Neumann norms of the Hardy-Steklov operator in Lebesgue function spaces on the semi-axis.
Recently, Hartz proved that every commuting contractive classical multishift with non-zero weights satisfies the matrix-version of von Neumann's inequality. We show that this result does not extend to the class of commuting operator-valued multishifts with invertible operator weights. In particular, we show that if $A$ and $B$ are commuting contractive $d$-tuples of operators such that $B$ satisfies the matrix-version of von Neumann's inequality and $(1, \ldots, 1)$ is in the algebraic spectrum of $B$, then the tensor product $A \otimes B$ satisfies the von Neumann's inequality if and only if …
A note on approximate amenability of type I von Neumann algebras
本文运用Ozawa与Runde的方法，证明了I型冯·诺依曼代数的近似顺从性与顺从性等价。
There are discussed some modifications of n-th von Neumann-Jordan constant considered by M. Kato, Y. Takahashi and K. Hashimoto. We study, among others, upper, lower, upper modified and lower modified n-th von Neumann-Jordan constant and relationships between them. There are characterized uniformly non-l_{n}^{1} Banach spaces in terms of the upper modified n-th von Neumann-Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces L^{p} and l^{p} (1\leq p\leq oo).
We will discuss some results of the paper "Asymptotic estimates on the Von Neumann Inequality for homogeneous polynomials", of Galicer D., Muro S. and Sevilla-Peris P. Also, we will see how to extend some of these results using the same techniques in such paper.
In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the free product of m>2 groups, all finite and having the same order p>2 (m>p-1). We prove actually more, namely that the unique trace preserving conditional expectations from these group factors onto their Laplacian subalgebras are asymptotic homomorphisms, which in turn implies that the subalgebras are strongly singular masas.
Von Neumann entropy of the angle operator between a pair of intermediate subalgebras
本文研究简单 C*-代数单位包含中一对中间子代数之间的夹角算子，并针对一大类四元组给出了其对偶夹角算子冯·诺依曼熵的可计算公式。
The category of von Neumann correspondences from B to C (or von Neumann B-C-modules) is dual to the category of von Neumann correspondences from C' to B' via a functor that generalizes naturally the functor that sends a von Neumann algebra to its commutant and back. We show that under this duality, called commutant, Rieffel's Eilenberg-Watts theorem (on functors between the categories of representations of two von Neumann algebras) switches into Blecher's Eilenberg-Watts theorem (on functors between the categories of von Neumann modules over two von Neumann algebras) and back.
We explore a duality between von-Neumann's mean ergodic theorem in von-Neumann algebra and Birkhoff's mean ergodic theorem in the pre-dual Banach space of von-Neumann algebras. Besides improving known mean ergodic theorems on von-Neumann algebras, we prove Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space.
Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we characterize lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, we give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
This paper is devoted to local derivations on subalgebras on the algebra $S(M, τ)$ of all $τ$-measurable operators affiliated with a von Neumann algebra $M$ without abelian summands and with a faithful normal semi-finite trace $τ.$ We prove that if $\mathcal{A}$ is a solid $\ast$-subalgebra in $S(M, τ)$ such that $p\in \mathcal{A}$ for all projection $p\in M$ with finite trace, then every local derivation on the algebra $\mathcal{A}$ is a derivation. This result is new even in the case standard subalgebras on the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H.$ We also ap…
Richard Kadison showed that not every commutative von Neumann subalgebra of a factor von Neumann algebra is equal to its relative double commutant. We prove that every commutative C*-subalgebra of a centrally prime C*-algebra $B$ equals its relative approximate double commutant. If $B$ is a von Neumann algebra, there is a related distance formula.
We prove that the von Neumann algebras generated by $n$ $q$-Gaussian elements, are factors for $n\ge 2$.
We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.
An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.
We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.
We studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
Let $M$ be a finite von Neumann algebra. In the first part, we give asymptotic results about $M$-stable sequences of weak*-continuous mappings which are related with operators belonging to $M$. In the second part, we extend, by a shorter way, similarity results given in [CaFa2] to unbounded semigroups of operators contained in a finite von Neumann algebra.
Additive spectrum preserving mappings from von Neumann algebras
本文证明了Jafarian的猜想，即从冯·诺伊曼代数到半单巴拿赫代数的加性谱保持映射必然是若尔当同构。
The aim of this paper is to study the logarithmic submajorisations inequalities for operators in a finite von Neumann algebra. Firstly, some logarithmic submajorisations inequalities due to Garg and Aulja are extended to the case of operators in a finite von Neumann algebra. As an application, we get some new Fuglede-Kadison determinant inequalities of operators in that circumstance. Secondly, we improve and generalize to the setting of finite von Neumann algebras, a generalized Hölder type generalized singular numbers inequality.
We prove that the group-measure-space von Neumann algebra $L^\infty(T^2) \rtimes SL(2,Z)$ is solid. The proof uses topological amenability of the action of $SL(2,Z)$ on the Higson corona of $Z^2$.
We introduce the fundamental group $F(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space. The form of $F(\mathcal{M})$ is completely determined. Moreover, there exists a finite von Neumann algebra $\mathcal{M}$ with finite dimensional normal trace space such that $F(\mathcal{M})=G$ for any conceivable groups $G$ as fundamental group.
We study the $t$-deformation of gaussian von Neumann algebras. They appear as example in the theories of Interacting Fock spaces and conditionally free products. When the number of generators is fixed, it is proved that if $t$ sufficiently close to 1, then these algebras do not depend on $t$. In the same way, the notion of conditionally free von Neumann algebras often coincides with freeness.
The goal of the paper is to study a topology generated by the star order on von Neumann algebras. In particular, it is proved that the order topology under investigation is finer than $σ$-strong* topology. On the other hand, we show that it is comparable with the norm topology if and only if the von Neumann algebra is finite-dimensional.
We investigate the Schatten-von Neumann properties of pseudo-differential operators using the method proposed by A. Boulkhemair in \cite{Boulkhemair 2}. The symbols are elements of the ideals $S_{w}^{p}$ of the Sjöstrand algebra $S_{w}$.
Let ${\mathcal M}$ be a von Neumann algebra with no central summands of type I$_1$. It is shown that every nonlinear $ξ-$Lie derivation ($ξ\neq 1$) on $\mathcal M$ is an additive derivation.
In this paper, I introduce the concept of $ (\upvarphi, \uppsi) $-finite von Neumann algebras and I show that if $ \mathscr{M} $ is a finite and $ (\upvarphi, \uppsi) $-finite von Neumann algebra togather with condition $ \{ \big( Δ(u+v)-Δ(u)-Δ(v)\big)^{*}\} \subseteq \uppsi(\mathscr{M}) $, then each (approximately) 2-local $ (\upvarphi, \uppsi) $-derivation $ δ$ on $ \mathscr{M} $, is a $ (\upvarphi, \uppsi) $-derivation.
We give an explicit formula for the time projection in an arbitrary von Neumann algebra from which all its basic properties can be easily derived. The analysis of the situation when this time projection is a conditional expectation is also performed.
The eigenvectors of an ergodic semigroup of linear normal positive unital maps on a von Neumann algebra are described. Moreover, it is shown by means of examples, that mere positivity of the maps in question is not sufficient for Frobenius theory as in S. Albeverio and R. Høegh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \bf{64}(1978), 83--94, to hold.
Recently, M. Daws introduced a notion of co-representation of abelian Hopf--von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf--von Neumann algebras. The key is our observation that, for a von Neumann algebra $\M$ and a reflexive operator space $E$, the normal spatial tensor product $\M \bar{\tensor} \CB(E)$ is a Banach algebra if and only if $\M$ is subhomogeneous or $E$ is completely isomorphic to column Hilbert space.
We prove that the predual of any von Neumann algebra is $1$-Plichko, i.e., it has a countably $1$-norming Markushevich basis. This answers a question of the third author who proved the same for preduals of semifinite von Neumann algebras. As a corollary we obtain an easier proof of a result of U.~Haagerup that the predual of any von Neumann algebra enjoys the separable complementation property. We further prove that the self-adjoint part of the predual is $1$-Plichko as well.
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $n\in \mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
On the amenable subalgebras of group von Neumann algebras
本文从动力学角度研究可数群的群冯·诺依曼代数的子代数，证明极大不变顺从子代数的存在性，并引入不变概率测度概念，证明顺从测度支撑于该极大子代数。
In this paper we consider permutations of sequences of partitions, obtaining a result which parallels von Neumann's theorem on permutations of dense sequences and uniformly distributed sequences of points.
On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
该研究引入群von Neumann代数酉元相关的测度收缩概念，分析迹交叉乘积von Neumann代数的子代数相对交换子，并将结果应用于负曲率群与SL(d,Z)（d≥2）。
In this talk, I will survey recent progress made on the classification of von Neumann algebras arising from countable groups and their actions on probability spaces. In particular, I will present the first results which provide classes of (superrigid) groups and actions that can be entirely reconstructed from their von Neumann algebras.
The heat semigroup on discrete hypercubes is well-known to be contractive over $L_p$-spaces for $1<p<\infty$. A question of Mendel and Naor \cite{MN14} concerns a stronger contraction property in the tail spaces, which is known as the heat-smoothing conjecture. Eskenazis and Ivanisvili \cite{EI20} considered a Gaussian analog of this conjecture and resolved some special cases. In particular, they proved that heat-smoothing type conjecture holds for holomorphic functions in the Gaussian spaces with sharp constants. In this paper, we prove analogous sharp inequalities for holomorphic subalgebras…
We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W$^*$-superrigid) actions that can be completely recovered from their von Neumann algebras and II$_1$ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.
In this paper, we present an infinitesimal version of the Stone-von Neumann Theorem. This work was motivated by the need to formulate the uniqueness property of the Heisenberg Commutation Relation purely in terms of unbounded operators.
We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup with norm-bounded generator on a von Neumann algebra equipped with a faithful semifinite trace. For von Neumann algebras with abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, H(phi)egh-Krohn and Olsen [Alb] for the special case of the finite dimensional ma…
The dynamical quantum Zeno effect is studied in the context of von Neumann algebras. We identify a localized subalgebra on which the Zeno dynamics acts by automorphisms. The Zeno dynamics coincides with the modular dynamics of that subalgebra, if an additional assumption is satisfied. This relates the modular operator of that subalgebra to the modular operator of the original algebra by a variant of the Kato-Lie-Trotter product formula.
In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.
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