Let E be a product system of C*-correspondences over N^r. Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of E are established and difference between regular and *-regular dilations discussed. It is in particular shown that a minimal isometric dilation is *-regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail.
Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend…
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.
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 …
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…
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 $\mathbb{R}_+^k$.
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.
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…
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…
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 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 p-adic magic contraction on p-adic Hilbert space. We derive p-adic Halmos dilation, p-adic Egervary N-dilation, p-adic von Neumann inequality and p-adic Sz.-Nagy dilation for p-adic magic contraction.
The explicit constructions of minimal isometric, and minimal unitary dilations of an arbitrary linear pencil of operators $T(λ)=T_0+λT_1$ consisting of contractions on a separable Hilbert space for $|λ|=1$, which generalize the classical constructions (the case $T_1=0$), are presented. In contrast to the classical case these dilations are essentially non-unique.
We develop a dilation theory for row contractions subject to constraints determined by sets of noncommutative polynomials. Under natural conditions on the constraints, we have uniqueness for the minimal dilation. A characteristic function is associated with any (constrained) row contraction. For pure constrained row contraction, we show that the characteristic function is a complete unitary invariant and provide a model. We also show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction can be express…
We study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.
We establish the existence and uniqueness of finite free resolutions - and their attendant Betti numbers - for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example, every graded d-contraction that is finitely multi-cyclic has a unique free cover of finite type - whose kernel is a Hilbert module inheriting the same properties. This contrasts sharply with what ca…
We show by a counter example the failure of rational dilation on the tetrablock, a polynomially convex and non-convex domain in $\mathbb C^3$, defined as $$ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever } |z|\leq 1, |w|\leq 1 \}. $$ A commuting triple of operators $(T_1,T_2,T_3)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called an $\mathbb E$-contraction. For an $\mathbb E$-contraction $(T_1,T_2,T_3)$, the two operator equations $$ T_1-T_2^*T_3=D_{T_3}X_1D_{T_3} \textup{ and } T_2-T_1^*T_3= D_{T_3}X_2D_{T_3}, \quad …
We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper.
The closed symmetrized tridisc $Γ_3$ and its distinguished boundary $bΓ_3$ are the sets $Γ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $bΓ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq Γ_3.$ A triple of commuting operators $(S_1,S_2,P)$ defined on a Hilbert space $\mathcal H$ for which $Γ_3$ is a spectral set is called a $Γ_3$-contraction. In this article we show by a counter example that there are $Γ_3$-contractions which do not dilate. It is also shown that under certain conditions a $Γ_3$-contraction…
An analysis is given of $*$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras $\A_θ$ and $\A_u$ which are associated with the commutation relation permutation $θ$ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix $u$ in $M_m(\bC) \otimes M_n(\bC)$. We show that a defect free row contractive representation has a unique minimal dilation to a $*$-representation and we provide a new simpler proof of Solel's row isometric dilation of two $u$-commuting row contractions. Furthermore it is shown that th…
A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ Γ_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n \right \} \] is a spectral set, is called a $Γ_n$-\textit{contraction}. A $Γ_n$-contraction is said to be \textit{pure} or $C._0$ if $P$ is $C._0$, that is, if ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. We show that for any $Γ_n$-contraction $(S_1,\dots, S_{n-1},P)$, there is a unique operator tuple $(A_1…
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 $X$.
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 take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. Th…
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).
In this paper, we establish the one-sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative $L_p$-spaces for a fixed $1<p<\infty$, which particularly applies to positive isometries and general positive Lamperti contractions or power bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces $L_p([0,1])$. Our study falls into neither the category of positive contractions considered by Junge-Xu nor the class of power bounded positive invertible operators considered by Hong…
Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call $Λ$-contractions or $Λ$-coisometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases.
We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications t…
Let $\mathbb{D}$ denote the unit disc in the complex plane $\mathbb{C}$ and let $\mathbb{D}^2 = \mathbb{D} \times \mathbb{D}$ be the unit bidisc in $\mathbb{C}^2$. Let $(T_1, T_2)$ be a pair of commuting contractions on a Hilbert space $\mathcal{H}$. Let $\mbox{dim } \mbox{ran}(I_{\mathcal{H}} - T_j T_j^*) < \infty$, $j = 1, 2$, and let $T_1$ be a pure contraction. Then there exists a variety $V \subseteq \overline{\mathbb{D}}^2$ such that for any polynomial $p \in \mathbb{C}[z_1, z_2]$, the inequality \[ \|p(T_1,T_2)\|_{\mathcal{B}(\mathcal{H})} \leq \|p\|_V \] holds. If, in addition, $T_2$ i…
When S is a discrete subsemigroup of a discrete group G such that G = S^{-1} S, it is possible to extend circle-valued multipliers from S to G; to dilate (projective) isometric representations of S to (projective) unitary representations of G; and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilate…
Let $A$ be a unital $C^*$-algebra and $α$ be an injective, unital endomorphism of $A$. A covariant representation of $(A,α)$ is a pair $(π,T)$ consisting of a $C^*$-representation $π$ of $A$ on a Hilbert space $H$ and a contraction $T$ in $B(H)$ satisfying $Tπ(α(a))=π(a)T$. It follows from more general results of ours that such a covariant representation can be extended to a covariant representation $(ρ,V)$ (on a larger space $K$) such that $V$ is a coisometry and it can be dilated to a covariant representation $(σ,U)$ (on a larger space $K_1$) with $U$ unitary. Our objective here is to give s…
The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp-semigroups and pairs (X,T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp-semigroups. Results we obtain: (i) a *-automorphic dilation to semigroups of *-endomorphisms over quite general semigroups; (ii) necessary and sufficient c…
Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\Sc{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\B{R}_+$ containing 0. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant represe…
