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.
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.
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. H\{o}egh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \textbf{64} (1978), 83--94, to hold.
We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative $\mathrm{L}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results implies the boundedness of the McIntosh's $\mathrm{H}^\infty$ fun…
Consider $d$ commuting $C_{0}$-semigroups (or equivalently: $d$-parameter $C_{0}$-semigroups) over a Hilbert space for $d \in \mathbb{N}$. In the literature (\textit{cf.} [29, 26, 27, 23, 18, 25]), conditions are provided to classify the existence of unitary and regular unitary dilations. Some of these conditions require inspecting values of the semigroups, some provide only sufficient conditions, and others involve verifying sophisticated properties of the generators. By focussing on semigroups with bounded generators, we establish a simple and natural condition on the generators, \textit{viz…
In this paper, we introduce Frobenius von Neumann algebras and study quantum convolution inequalities. In this framework, we unify quantum Young's inequality on quantum symmetries such as subfactors, and fusion bi-algebras studied in quantum Fourier analysis. Moreover, we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case. We also prove quantum smooth entropic convolution inequalities. We obtain the positivity of comultiplications of subfactor planar algebras, which is stronger than the quantum Schur product theorem. All these inequalities pr…
The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state $φ$ is shown to imply existence of unital, $φ$-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms.
In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.
We generalize some basic C*-algebra and von Neumann algebra theory on hereditary C*-subalgebras and projections. In particular, we extend Murray-von Neumann equivalence from projections to *-annihilators and show that several of its important properties also extend from von Neumann algebras to proper *-semigroups. Furthermore, we show how to obtain *-annihilator decompositions of a proper *-semigroup that correspond to classical von Neumann algebra type decompositions.
Recent results of L. Zsido, based on his previous work with C. P. Niculescu and A. Stroh, on actions of topological semigroups on von Neumann algebras, give a Jacobs-de Leeuw-Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf-von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.
It is known that every semigroup of normal completely positive maps of a von Neumann can be ``dilated" in a particular way to an E_0-semigroup acting on a larger von Neumann algebra. The E_0-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of {\it minimality} are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is $\Cal B(H)$. In this paper we clarify these issues by giving a new char…
We prove that $(λ_g\mapsto e^{-t|g|^r}λ_g)_{t>0}$ defines a completely bounded semigroup of multipliers on the von Neuman algebra of hyperbolic groups for all real number $r$. One ingredient in the proof is the observation that a construction of Ozawa allows to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup--Steenstrup--Szwarc and Wysoczański. Another ingredient is an upper estimate of trace class norms for Hankel matrices, which is based on Peller's characterization of such norms.
Let $\{ϕ_s\}_{s\in S}$ be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra $N$. Assume there exists a semigroup $\{α_s\}_{s\in S}$ of weak*-continuous *-endomorphisms of some larger von Neumann algebra $M\supset N$ and a projection $p\in M$ with $N=pMp$ such that $α_s(1-p)\le 1-p$ for every $s\in S$ and $ϕ_s(y)=pα_s(y)p$ for all $y\in N$. If $\inf_{s\in S}α_s(1-p)=0$ then we show that the map $E:M\to N$ defined by $E(x)=pxp$ for $x\in M$ induces a complete isometry between the fixed point spaces of $\{α_s\}_{s\in S}$ …
Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^*$-algebra $C^*_δ(S)$ generated by the operators of translation by all elements of $S$ in $L^2(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. The same is proved for the von Neumann algebra $VN(S)$ generated by $C^*_δ(S)$.
We study completely contractive representations of product systems of $C^*$-correspondences over semigroups. For a product system of $C^*$-correspondences over the semigroup $\mathbb{N}^2$, we prove that every such representation can be dilated to an isometric (or Toeplitz) representation. We use it to 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).
Borchers and Wiesbrock have demonstrated certain results concerning the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables. These results are abstracted and analyzed as essentially operator-theoretic. Criteria are then demonstrated for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms. All this is combined to establish a von Neumann-algebraic converse to the Borchers-Wiesbrock results. This analysis is then applied to questions of isotony and co…
Two different models for a Hopf-von Neumann algebra of bounded functions on the quantum semigroup of all (quantum) permutations of infinitely many elements are proposed, one based on projective limits of enveloping von Neumann algebras related to finite quantum permutation groups, and the second on a universal property with respect to infinite magic unitaries.
We give sufficient conditions on an operator space $E$ and on a semigroup of operators on a von Neumann algebra $M$ to obtain a bounded analytic or a $R$-analytic semigroup $(T_t \otimes Id_E)_{t \geq 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^\infty(Σ_θ)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.
We study asymptotic behavior of a Markov semigroup on a von-Neumann algebra by exploring a maximal von-Neumann subalgebra where the Markov semigroup is an automorphism. This enables us to prove that strong mixing is equivalent to ergodic property for continuous time Markov semigroup on a type-I von-Neumann algebra with center completely atomic. For discrete time dynamics we prove that an aperiodic ergodic Markov semigroup on a type-I von-Neumann algebra with center completely atomic is strong mixing. There exists a tower of isomorphic von-Neumann algebras generated by the weak Markov process a…
Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint is a bimodule map. In this note we prove that they split as an orthogonal direct sum of two closed ideals, their Jacobson radical which in fact is nothing but their annihilator, and the closure of the linear span of their group-like elements. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In par…
Product systems have been originally introduced to classify E$_0$-semigroups on type I factors by Arveson. We develop the classification theory of E$_0$-semigroups on a general von Neumann algebra and the dilation theory of CP$_0$-semigroups in terms of W$^*$-bimodules. For this, we provide a notion of product system of W$^*$-bimodules. This is a W$^*$-bimodule version of Arveson's and Bhat-Skeide's product systems. There exists a one-to-one correspondence between CP$_0$-semigroups and units of product systems of W$^*$-bimodules. The correspondence implies a construction of a dilation of a giv…
We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm{VN}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. Our construction relies on probabilistic tools and is even new for the group $\mathbb{R}^n$. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm{L}^p$-spaces.
We consider semigroups of operators on a W$^*$-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a "stable" and "reversible" part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W$^*$-algebras.
We prove that for every semigroup of Schwarz maps on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak$^*$ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe th…
We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.
In a previous paper, we showed that every strongly commuting pair of CP_0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E_0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter…
In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.
We review some of our results from the theory of product systems of Hilbert modules. We explain that the product systems obtained from a CP-semigroup in a paper by Bhat and Skeide and in a paper by Muhly and Solel are commutants of each other. Then we use this new commutant technique to construct product systems from E_0-semigroups on B^a(E) where E is a strongly full von Neumann module. (This improves the construction from a paper by Skeide for Hilbert modules where existence of a unit vector is required.) Finally, we point out that the Arveson system of a CP-semigroup constructed by Powers f…
We give a necessary and sufficient criterion when a normal CP-map on a von Neumann algebra admits a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on B(G).
We prove that if $\{ϕ_t\}_{t \geq 0}$ is a CP-semigroup acting on a von Neumann algebra $M \subseteq B(H)$, then for every $A\in M$ and $ξ\in H$, the map $t \mapsto ϕ_t(A)ξ$ is norm-continuous. We discuss the implications of this fact to the existence of dilations of CP-semigroups to semigroups of endomorphisms.
A classical result of Sherman says that if the space of self-adjoint elements in a $C^*$-algebra $\mathcal{A}$ is a lattice with respect to its canonical order, then $\mathcal{A}$ is commutative. We give a new proof of this theorem which shows that it is intrinsically connected with the spectral theory of positive operator semigroups. Our methods also show that some important Perron--Frobenius like spectral results fail to hold in any non-commutative $C^*$-algebra.
W. Paschke's version of Stinespring's theorem associates a Hilbert $C^*$-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a $C^*$-algebra $\mathcal A$ one may associate an inclusion system $E=(E_t)$ of Hilbert $\mathcal A$-$\mathcal A$-modules with a generating unit $ξ=(ξ_t)$. Suppose $\mathcal B$ is a von Neumann algebra, consider $M_2(\mathcal B)$, the von Neumann algebra of $2\times 2$ matrices with entries from $\mathcal B$. Suppose $(Φ_t)_{t\ge 0}$ with $Φ_t=\begin{pmatrix} ϕ_t^1& ψ_t ψ_t^*&ϕ_t^2 \end{p…
Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C$^*$-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.
This paper presents the complete classification of E_0-semigroups by product systems in the case of von Neumann correspondences, and under countability assumptions in the case of C*-correspondences.
Inspired by the construction of the Cuntz semigroup for a C*-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C*-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimens…
In this paper we consider a semigroup of completely positive maps $τ=(τ_t,t \ge 0)$ with a faithful normal invariant state $ϕ$ on a type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve this via a more general Kolmogorov's type of construction for stationary Markov processes which naturally associate a nested isomorphic von-Neumann algebras. In particular this construction generalizes well known Jones construction associated with a sub-factor of type-II$_1$ factor.
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.
