Commutative Von Neumann Algebra Research Articles

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Gelfand-type duality for commutative von Neumann algebras

We show that the following five categories are equivalent: (1) the opposite category of commutative von Neumann algebras; (2) compact strictly localizable enhanced measurable spaces; (3) measurable locales; (4) hyperstonean locales; (5) hyperstonean spaces. This result can be seen as a measure-theoretic counterpart of the Gelfand duality between commutative unital C*-algebras and compact Hausdorff topological spaces.

Quantum incompatibility of channels with general outcome operator algebras

A pair of quantum channels is said to be incompatible if they cannot be realized as marginals of a single channel. This paper addresses the general structure of the incompatibility of completely positive channels with a fixed quantum input space and with general outcome operator algebras. We define a compatibility relation for such channels by identifying the composite outcome space as the maximal (projective) C*-tensor product of outcome algebras. We show theorems that characterize this compatibility relation in terms of the concatenation and conjugation of channels, generalizing the recent result for channels with quantum outcome spaces. These results are applied to the positive operator valued measures (POVMs) by identifying each of them with the corresponding quantum-classical (QC) channel. We also give a characterization of the maximality of a POVM with respect to the post-processing preorder in terms of the conjugate channel of the QC channel. We consider another definition of compatibility of normal channels by identifying the composite outcome space with the normal tensor product of the outcome von Neumann algebras. We prove that for a given normal channel, the class of normally compatible channels is upper bounded by a special class of channels called tensor conjugate channels. We show the inequivalence of the C*- and normal compatibility relations for QC channels, which originates from the possibility and impossibility of copying operations for commutative von Neumann algebras in C*- and normal compatibility relations, respectively.

On the image, characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ ( σ , τ )-derivations

The first main theorem of this paper asserts that any $$(\sigma , \tau )$$ -derivation d, under certain conditions, either is a $$\sigma $$ -derivation or is a scalar multiple of ( $$\sigma - \tau $$ ), i.e. $$d = \lambda (\sigma - \tau )$$ for some $$\lambda \in \mathbb {C} \backslash \{0\}$$ . By using this characterization, we achieve a result concerning the automatic continuity of $$(\sigma , \tau $$ )-derivations on Banach algebras which reads as follows. Let $$\mathcal {A}$$ be a unital, commutative, semi-simple Banach algebra, and let $$\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}$$ be two distinct endomorphisms such that $$\varphi \sigma (\mathbf e )$$ and $$\varphi \tau (\mathbf e )$$ are non-zero complex numbers for all $$\varphi \in \Phi _\mathcal {A}$$ . If $$d : \mathcal {A} \rightarrow \mathcal {A}$$ is a $$(\sigma , \tau )$$ -derivation such that $$\varphi d$$ is a non-zero linear functional for every $$\varphi \in \Phi _\mathcal {A}$$ , then d is automatically continuous. As another objective of this research, we prove that if $$\mathfrak {M}$$ is a commutative von Neumann algebra and $$\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$$ is an endomorphism, then every Jordan $$\sigma $$ -derivation $$d:\mathfrak {M} \rightarrow \mathfrak {M}$$ is identically zero.

Characterization of some derivations on von Neumann algebras via left centralizers

Let \(\mathfrak {M}\) be a von Neumann algebra, and let \(\mathfrak {T}:\mathfrak {M} \rightarrow \mathfrak {M}\) be a bounded linear map satisfying \(\mathfrak {T}(P^{2}) = \mathfrak {T}(P)P + \Psi (P,P)\) for each projection P of \(\mathfrak {M}\), where \(\Psi :\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is a bi-linear map. If \(\Psi \) is a bounded l-semi Hochschild 2-cocycle, then \(\mathfrak {T}\) is a left centralizer associated with \(\Psi \). By applying this conclusion, we offer a characterization of left \(\sigma \)-centralizers, generalized derivations and generalized \(\sigma \)-derivations on von Neumann algebras. Moreover, it is proved that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every bi-\(\sigma \)-derivation \(D:\mathfrak {M} \times \mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.

Topos quantum theory reduced by context-selection functors

In this paper, we deal with quantum theories on presheaves and sheaves on context categories consisting of commutative von Neumann algebras of bounded operators on a Hilbert space, from two viewpoints. One is to reduce presheaf-based topos quantum theory via sheafification, and the other is to import quantum probabilities to the reduced sheaf quantum theory. The first is done by means of a functor that selects some expedient contexts. It defines a Grothendieck topology on the category consisting of all contexts, hence, induces a sheaf topos on which we construct a downsized quantum theory. Also, we show that the sheaf quantum theory can be replaced by an equivalent, more manageable presheaf quantum theory. Quantum probabilities are imported by means of a Grothendieck topology that is defined on a category consisting of probabilities and enables to regard them as intuitionistic truth-values. From these topologies, we construct another Grothendieck topology that is defined on the product of the context category and the probability category and reflects the selection of contexts and the identification of probabilities with truth-values. We construct a quantum theory equipped with quantum probabilities as truth-values on the sheaf topos induced by the Grothendieck topology.

Generalized derivations on commutative Von Neumann algebras

The main purpose of this paper is to prove that every generalized derivation on a commutative von Neumann algebra is identically zero.

Algebraic Davis Decomposition and Asymmetric Doob Inequalities

In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let $(\M,\tau)$ be a noncommutative probability space equipped with a weak-$*$ dense filtration of von Neumann subalgebras $(\M_n)_{n \ge 1}$. Let $\E_n$ denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for $1 < p < 2$ and $x \in L_p(\M,\tau)$ one can find $a, b \in L_p(\M,\tau)$ and contractions $u_n, v_n \in \M$ such that $$\E_n(x) = a u_n + v_n b \quad \mbox{and} \quad \max \big\{ \|a\|_p, \|b\|_p \big\} \le c_p \|x\|_p.$$ Moreover, it turns out that $a u_n$ and $v_n b$ converge in the row/column Hardy spaces $\H_p^r(\M)$ and $\H_p^c(\M)$ respectively. In particular, this solves a problem posed by Defant and Junge in 2004. In the case $p=1$, our results establish a noncommutative form of Davis celebrated theorem on the relation between martingale maximal and square functions in $L_1$, whose noncommutative form has remained open for quite some time. Given $1 \le p \le 2$, we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of $\E_n(x)$ in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.

Duality, convexity and peak interpolation in the Drury–Arveson space

We consider the closed algebra Ad generated by the polynomial multipliers on the Drury–Arveson space. We identify Ad⁎ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop–Carleson–Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light on the nature of the extreme points of the unit ball of Ad⁎.

Topos quantum theory on quantization-induced sheaves

In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheaf-based one can be constructed on sheaves defined by the quantization-induced Lawvere-Tierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheaf-based expressions of physical propositions and truth objects, and thereby give a method of truth-value assignment to the propositions. Furthermore, we clarify the relationship to the presheaf-based quantum theory. We give translation rules between the sheaf-based ingredients and the corresponding presheaf-based ones. The translation rules have “coarse-graining” effects on the spaces of the presheaf-based ingredients; a lot of different proposition presheaves, truth presheaves, and presheaf-based truth-values are translated to a proposition sheaf, a truth sheaf, and a sheaf-based truth-value, respectively. We examine the extent of the coarse-graining made by translation.

Geometric description of the preduals of atomic commutative von Neumann algebras

Strongly facially symmetric spaces isometrically isomorphic to the predual space of an atomic commutative von Neumann algebra are described

On notion of asymptotic derivations

AbstractIn this paper we introduce some notion of asymptotic derivations of a C*‐ and W*‐dynamical systems which naturally arises from a one‐parameter group of automorphisms. We show that an asymptotic derivation on a unital simple C*‐algebra or a von Neumann algebra is asymptotically inner. Every asymptotic derivation on finite dimensional C*‐algebras is induced by an inner derivation. We prove that any asymptotic derivation on commutative von Neumann algebras have a strong limit zero. An asymptotic derivation on the hyperfinite II1‐factor given by some one‐parameter group of automorphisms can be induced by a (inner) derivation. Finally, we show that every asymptotic Jordan derivation of a C*‐dynamical system is an asymptotic derivation and is also induced by a derivation if there exists a limit.

LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

Infinite-Dimensional Representations of 2-Groups

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners - features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and intertwiners. We also study "irretractable" representations - another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C 0(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.

On unitary representability of topological groups

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

Maximality of Positive Operator-valued Measures

A positive operator-valued measure is a (weak-star) countably additive set function from a σ-field Σ to the space of nonnegative bounded operators on a separable complex Hilbert space Open image in new window . Such functions can be written as M = V*E(·)V in which E is a spectral measure acting on a complex Hilbert space Open image in new window and V is a bounded operator from Open image in new window to Open image in new window such that the only closed linear subspace of Open image in new window , containing the range of V and reducing E (Σ), is Open image in new window itself. Attention is paid to an existing notion of maximality for positive operator-valued measures. The purpose of this paper is to show that M is maximal if and only if E, in the above representation of M, generates a maximal commutative von Neumann algebra.

Analysis of the affine transformations of the time-frequency plane

We consider two aspects of the action of the extended metaplectic representation of the groupGof affine, measure and orientation preserving maps of the time-frequency plane onL2functions on the line. On the one hand, we list, up to equivalence, all possible reproducing formulas that arise by restricting the representation to connected Lie subgroups ofG.On the other hand, we describe, in terms of Weyl calculus, the commutative von Neumann algebras generated by restriction to one-parameter subgroups.

Arens algebras, associated with commutative von Neumann algebras

Arens algebras, associated with commutative von Neumann algebras

$L^p$-spaces and quantum dynamical semigroups

Introduction. The purpose of the present note is to show the role played by noncommutative L-spaces in the theory of quantum dynamical semigroups. We consider both the C∗-algebra and von Neumann algebra case, concentrating on the latter. The two cases are very different, a phenomenon easy to explain on the grounds of noncommutative measure theory. If we take a locally compact space with a Radon measure, then the isomorphism class of the corresponding L-spaces (p 6= 2) depends crucially on the choice of the measure. It is therefore only natural to expect the isomorphism class of L-spaces associated with a noncommutative C∗-algebra to depend on the choice of a weight (or state) on the algebra. This is further supported by the results of the final section of the paper where the natural definition of L-spaces for UHF algebras leads to such a dependence. On the other hand, two isomorphic von Neumann algebras lead to linearly isometric L-spaces and that does not depend on the choice of (faithful) weights on the algebras. This corresponds to the classical fact that two equivalent measures on a measurable space give rise to isomorphic L-spaces. Note that a commutative von Neumann algebra corresponds to a quasi-measure space, i.e. a measure space with an associated