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).
Here we study dilations of q-commuting tuples. In [BBD] the authors gave the correspondence between the two standard dilations of commuting tuples and here these results have been extended to q-commuting tuples. We are able to do this when q-coefficients `$q_{ij}$' are of modulus one. We introduce `maximal q-commuting subspace ' of a $n$-tuple of operators and `standard q-commuting dilation'. Our main result is that the maximal q-commuting subspace of the standard noncommuting dilation of q-commuting tuple is the `standard q-commuting dilation'. We also introduce q-commuting Fock space as the …
In this note we prove that there exist at least two examples of three commuting, unital, completely positive maps that have no dilation on a type I factor, and no minimal dilation on any von Neumann algebra.
We study dilations of finite tuples of normal, completely positive and completely contractive maps (which we call CP-maps) acting on a von Neumann algebra, and commuting according to a graph G. We show that if G is acyclic, then a tuple commuting according to it has a simultaneous *-endomorphic dilation, which also commutes according to G. Conversely, if G has a cycle, we exhibit an example of a tuple of CP-maps commuting according to G, which does not have an *-endomorphic dilation commuting according to G. To achieve these results we use dilation theory of representations of subproduct syste…
Laca constructed a minimal automorphic dilation for every semigroup dynamical system arising from an action of an Ore semigroup by injective endomorphisms of a unital $C^*$-algebra. Here we show that the semigroup crossed product with its action by inner endomorphisms given by the implementing isometries has as minimal automorphic dilation the group crossed product of the original dilation. Applications include recent examples studied by Cuntz and the second named author.
We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP_0-semigroups on B(H), H finite dimensional, has an E_0-dilation.
We describe a Markov dilation for any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$.
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…
Dilations of completely positive semigroups to endomorphism semigroups have been studied by numerous authors. Most existing dilation theorems involve a non-unital embedding, corresponding to the embedding of $B(H)$ as a corner of $B(K)$ for Hilbert spaces $H \subset K$. A 1986 paper of Jean-Luc Sauvageot shows how to achieve a unital dilation, but does not specify how to do so while also preserving continuity properties of the original semigroup. This thesis further develops Sauvageot's dilation theory in order to establish the existence of continuous unital dilations, and to explore connectio…
We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction there are two commonly used dilations in multivariable operator theory. Firstly there is the minimal isometric dilation consisting of isometries with orthogonal ranges and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on Boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all rep…
Using ideas due to Jean-Luc Sauvageot, we prove the existence of a continuous unital dilation of a CP$_0$-semigroup on a separable W$^*$-algebra. This paper presents the material in the author's Ph. D. thesis (arXiv.org:1304.0134.pdf) with some generalizations and an improved exposition.
There are known three ways to construct the minimal dilation of the discrete semigroup generated by a normal unital completely positive map on a von Neumann algebra, which are given by Arveson, Bhat-Skeide and Muhly-Solel. In this paper, we clarify the relation of the constructions by Bhat-Skeide and Muhly-Solel.
This paper has been withdrawn by the author, because a better treatment is given in the author's Phd. thesis (Sections 3.4.6 and 4.4), now available on the arxiv.
We consider a family of Hecke C*-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.
This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C*-algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representat…
We show that every interaction group extending an action of an Ore semigroup by injective unital endomorphisms of a C*-algebra, admits a dilation to an action of the corresponding enveloping group on another unital C*-algebra, of which the former is a C*-subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed. If the action is induced by covering maps on the spectrum, then the expectation is faithful.
In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb{R}_+^k$.
Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $ϕ$ and $θ$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $α$ and $β$ such that $ϕ_s \circ θ_t (P_H A P_H) = P_H α_s \circ β_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $ϕ$ is not an automorphism semigroup then $α$ is cocycle conjugate to the minimal *-endomorphic dilation of $ϕ$, and that if $ϕ$ is an automorphism semigroup then $α$ is also an automorphism semigroup. In particular, we conclude that if $ϕ$ is not an automorphism semigroup a…
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.
In this paper, we construct, for a certain class of semigroup dynamical systems, two operator algebras that are universal with respect to their corresponding covariance conditions: one being self-adjoint, and another being non-self-adjoint. We prove that the $C^*$-envelope of the non-self-adjoint operator algebra is precisely the self-adjoint one. This result leads to a number of new examples of operator algebras and their $C^*$-envelopes, with many from number fields and commutative rings. We further establish the functoriality of these operator algebras along with their applications.
The weak operator topology closed operator algebra on $L^2(R)$ generated by the one-parameter semigroups for translation, dilation and multiplication by $exp(iλx), λ\geq 0$, is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, $A_{ph}$, is antisymmetric in the sense that $A_{ph} \cap A_{ph}^*= CI$, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is $R$. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalen…
We construct a weak dilation of a not necessarily unital CP-semigroup to an E-semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E-semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to proof that covariant representations of product systems admit isometric dilations.
Regular dilation has recently been extended to graph product of $\mathbb{N}$, where having a *-regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among *-regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for *-regular dilation.
We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three $3\times3$ commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.
We introduce the notion of $Q$-commuting operators which is a generalization of commuting operators. We prove a generalized version of commutant lifting theorem and Ando's dilation theorem in the context of $Q$-commuting operators.
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 present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations.
It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the inde…
A particular case of the fundamental Sz.-Nagy--Foias functional model for a contraction states that a pure contraction always dilates to a pure isometry. We are interested in the similar question for pairs, more precisely: does a pair of commuting pure contractions always dilate to a pair of commuting pure isometries? The purpose of this article is to identify pairs of commuting pure contractions for which the above question has an affirmative answer. Our method is based on an explicit structure of isometric dilation for pure pairs of commuting contractions obtained in a recent work by Das, Sa…
We show that whenever a contractive $k$-tuple $T$ on a finite dimensional space $H$ has a unitary dilation, then for any fixed degree $N$ there is a unitary $k$-tuple $U$ on a finite dimensional space so that $q(T) = P_H q(U) |_H$ for all polynomials $q$ of degree at most $N$.
We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0…
We prove the existence of Hudson Parthasarathy dilation of a quantum dynamical semigroup on $B(\clh),$ which is symmetric with respect to the canonical normal trace on it.
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…
This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If $a_1,...,a_k$ are $\ast$-free $\mathscr{R}$-diagonal operators in a $\mathrm{II}_1$ factor, then $D_t(a_{i_1}... a_{i_n}) = e^{-nt} a_{i_1}... a_{i_n}$ defines a dilation semigroup on the non-self-adjoint operator algebra generated by $a_1,...,a_k$. We show that $D_t$ extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by $a_1,...,a_k$. Moreover, we show that $D_t$ satisfies an optimal ultracontractive proper…
Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We always assume that $n \geq 3$. The importance of this class of $n$-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for $n$-tuples, $n \geq 3$, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, w…
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in \cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
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…
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on $\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stoc…
This thesis is dedicated to developing a dilation theory for semigroups of completely positive maps. The first part treats two-parameter semigroups, and contains also contributions to dilation theory of product system representations. The second part deals with completely positive semigroups parameterized by quite general semigroups, where the major technical tool introduced is subproduct systems and their representations. In the third part subproduct systems are studied, together with the multivariable operator theory and operator algebras they give rise to.
A contractive $n$-tuple $A=(A_1,...,A_n)$ has a minimal joint isometric dilation $S=(S_1,...,S_n)$ where the $S_i$'s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the \wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe s…
A numerical index is introduced for semigroups of completely positive maps of $\Cal B(H)$ which generalizes the index of E_0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E_0-semigroup, provided that the dilation is minimal.
We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V…
We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single ch…
The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von …
We study completely contractive representations of product systems $X$ of correspondences over the semigroup $\mathbb{Z}_+^k$. We present a necessary and sufficient condition for such a representation to have a regular isometric dilation. We discuss representations that doubly commute and show that these representations induce completely contractive representations of the norm closed algebra generated by the image of the Fock representation of $X$.
Given a contractive tuple of Hilbert space operators satisfying certain $A$-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximal $A$-relation piece. We define a maximal piece more generally for a finite set of polynomials in $n$ noncommuting variables. We classify all representations of Cuntz-Krieger algebras ${\mathcal O}_A$ obtained from dilations of commuting tuples satisfying $A$-relations. The…
A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extended a result due to Athavale to representations on abelian lattice ordered semigroups.
We extended the definition of regular dilation to graph products of $\mathbb{N}$, which is an important class of quasi-lattice ordered semigroups. Two important results in dilation theory are unified under our result: namely, Brehmer's regular dilation on $\mathbb{N}^k$ and Frazho-Bunce-Popescu's dilation of row contractions. We further show that a representation of a graph product has an isometric Nica-covariant dilation if and only if it is $\ast$-regular. A special case of our result was considered by Popescu, and we studied the connection with Popescu's work.
We explore aspects of dilation theory in the finite dimensional case and show that for a commuting $n$-tuple of operators $T=(T_1,...,T_n) $ acting on some finite dimensional Hilbert space $H$ and a compact set $X\subset \mathbb{C}^n$ the following are equivalent: 1. $T$ has a normal $ X$-dilation. 2. For any $m\in \mathbb{N}$ there exists some finite dimensional Hilbert space $K$ containing $H$ and a tuple of commuting normal operators $N=(N_1,...,N_n)$ acting on $K$ such that $$ q(T)=P_Hq(N)|_H$$ for all polynomials $q$ of degree at most $m$ and such that the joint spectrum of $N$ is contain…
