arXiv subjects

Baruch Solel

Publications and source records attributed to Baruch Solel.

At least 19 recordsLinked to original sources

arXiv2026

Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions

We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of $C^*$-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple $(\sigma, T_1, T_2, \ldots, T_n)$, where each $(\sigma,T_i)$ is an isometric covariant representation of a $C^*$\nobreakdash-correspondence. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of product system.

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arXiv2021

Weighted Cuntz-Krieger Algebras

Let $E$ be a finite directed graph with no sources or sinks and write $X_E$ for the graph correspondence. We study the $C^*$-algebra $C^*(E,Z):=\mathcal{T}(X_E,Z)/\mathcal{K}$ where $\mathcal{T}(X_E,Z)$ is the $C^*$-algebra generated by weighted shifts on the Fock correspondence $\mathcal{F}(X_E)$ given by a weight sequence $\{Z_k\}$ of operators $Z_k\in \mathcal{L}(X_{E^{k}})$ and $\mathcal{K}$ is the algebra of compact operators on the Fock correspondence. If $Z_k=I$ for every $k$, $C^*(E,Z)$ is the Cuntz-Krieger algebra associated with the graph $E$. We show that $C^*(E,Z)$ can be realized as a Cuntz-Pimsner algebra and use a result of Schweizer to find conditions for the algebra $C^*(E,Z)$ to be simple. We also analyse the gauge-invariant ideals of $C^*(E,Z)$ using a result of Katsura and conditions that generalize the conditions of subsets of $E^0$ (the vertices of $E$) to be hereditary or saturated. As an example, we discuss in some details the case where $E$ is a cycle.

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arXiv2021

Dilations of unitary tuples

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric \[ d_D(u,v)=\log\max\{c(u,v),c(v,u)\} \] on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_Θ$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{iθ_{k,\ell}} u_k u_\ell$, where $Θ=(θ_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_Θ, u_{Θ'})\leq e^{\frac{1}{4}\|Θ-Θ'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $Θ\mapsto U(Θ)\in B(H)^d$ such that $U(Θ)\sim u_Θ$ and \[ \|U(Θ)-U({Θ'})\|\leq K\|Θ-Θ'\|^{1/2}. \] Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] From this, we recover Passer's upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.

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arXiv2021

Weighted Cuntz Algebras

We study the $C^*$-algebra $\mathcal{T}/\mathcal{K}$ where $\mathcal{T}$ is the $C^*$-algebra generated by $d$ weighted shifts on the Fock space of $\mathbb{C}^d$, $\mathcal{F}(\mathbb{C}^d)$, ( where the weights are given by a sequence $\{Z_k\}$ of matrices $Z_k\in M_{d^k}(\mathbb{C})$) and $\mathcal{K}$ is the algebra of compact operators on the Fock space. If $Z_k=I$ for every $k$, $\mathcal{T}/\mathcal{K}$ is the Cuntz algebra $\mathcal{O}_d$. We show that $\mathcal{T}/\mathcal{K}$ is isomorphic to a Cuntz-Pimsner algebra and use it to find conditions for the algebra to be simple. We present examples of simple and of non simple algebras of this type. We also describe the $C^*$-representations of $\mathcal{T}/\mathcal{K}$.

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arXiv2020

Invariant subspaces for certain tuples of operators with applications to reproducing kernel correspondences

The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples $T=(T_1,\ldots,T_d)$ operators that are contractive and pure; that is $Φ_T(I)\leq I$ and $Φ_T^n(I)\searrow 0$ where $$Φ_T(a)=Σ_i T_iaT_i^*.$$ Here we generalize some of their results to commuting tuples $T$ satisfying similar conditions but for $$Φ_T(a)=Σ_{α\in \mathbb{F}^+_d} x_{|α|}T_αaT_α^*$$ where $\{x_k\}$ is a sequence of non negative numbers satisfying some natural conditions (where $T_α=T_{α(1)}\cdots T_{α(k)}$ for $k=|α|$). In fact, we deal with a more general situation where each $x_k$ is replaced by a $d^k\times d^k$ matrix. We also apply these results to subspaces of certain reproducing kernel correspondences $E_K$ (associated with maps-valued kernels $K$) that are invariant under the multipliers given by the coordinate functions.

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arXiv2017

Minimal and maximal matrix convex sets

To every convex body $K \subseteq \mathbb{R}^d$, one may associate a minimal matrix convex set $\mathcal{W}^{\textrm{min}}(K)$, and a maximal matrix convex set $\mathcal{W}^{\textrm{max}}(K)$, which have $K$ as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies $K,L \subseteq \mathbb{R}^d$ does $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$ hold? For a convex body $K$, we aim to find the optimal constant $θ(K)$ such that $\mathcal{W}^{\textrm{max}}(K) \subseteq θ(K) \cdot \mathcal{W}^{\textrm{min}}(K)$; we achieve this goal for all the $\ell^p$ unit balls, as well as for other sets. For example, if $\overline{\mathbb{B}}_{p,d}$ is the closed unit ball in $\mathbb{R}^d$ with the $\ell^p$ norm, then \[ θ(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{\textrm{max}}(K) = \mathcal{W}^{\textrm{min}}(K)$ if and only if $K$ is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every $d$-tuple of self-adjoint operators of norm less than or equal to $1$, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.

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arXiv2016

Dilations, inclusions of matrix convex sets, and completely positive maps

A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq 1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n \times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $\mathcal{S} = \cup_{n \geq 1} \mathcal{S}_n,$ and $\mathcal{T} = \cup_{n \geq 1} \mathcal{T}_n$, we find geometric conditions on $\mathcal{S}$ or on $\mathcal{T}$, such that $\mathcal{S}_1 \subseteq \mathcal{T}_1$ implies that $\mathcal{S} \subseteq C\mathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $\mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $\mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=\sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.

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arXiv2015

Boundaries, Bundles and Trace Algebras

We describe how noncommutative function algebras built from noncommutative functions in the sense of \cite{K-VV2014} may be studied as subalgebras of homogeneous $C^{*}$-algebras.

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arXiv2015

Matricial Function Theory and Weighted Shifts

Let $\mathcal{T}_{+}(E)$ be the tensor algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. In earlier work, we showed that the completely contractive representations of $\mathcal{T}_{+}(E)$, whose restrictions to $M$ are normal, are parametrized by certain discs or balls $\overline{D(E,σ)}$ indexed by the normal $*$-representations $σ$ of $M$. Each disc has analytic structure, and each element $F\in \mathcal{T}_{+}(E) $ gives rise to an operator-valued function $\widehat{F}_σ$ on $\overline{D(E,σ)}$ that is continuous and analytic on the interior. In this paper, we explore the effect of adding operator-valued weights to the theory. While the statements of many of the results in the weighted theory are anticipated by those in the unweighted setting, substantially different proofs are required. Interesting new connections with the theory of completely positive are developed. Our perspective has been inspired by work of Vladimir Müller in which he studied operators that can be modeled by parts of weighted shifts. Our results may be interpreted as providing a description of operator algebras that can be modeled by weighted tensor algebras. Our results also extend work of Gelu Popescu, who investigated similar questions.

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arXiv2012

Tensorial Function Theory: From Berezin transforms to Taylor's Taylor series and back

Let $H^{\infty}(E)$ be the Hardy algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. Then the ultraweakly continuous completely contractive representations of $H^{\infty}(E)$ are parametrized by certain sets $\mathcal{AC}(σ)$ indexed by $NRep(M)$ - the normal *-representations $σ$ of $M$. Each set $\mathcal{AC}(σ)$ has analytic structure, and each element $F\in H^{\infty}(E)$ gives rise to an analytic operator-valued function $\hat{F}_σ$ on $\mathcal{AC}(σ)$ that we call the $σ$-Berezin transform of $F$. The sets ${\mathcal{AC}(σ)}_{σ\inΣ}$ and the family of functions ${\hat{F}_σ}_{σ\inΣ}$ exhibit "matricial structure" that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.

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arXiv2011

Absolute continuity, Interpolation and the Lyapunov order

We extend our Nevanlinna-Pick theorem for Hardy algebras and their representations to cover interpolation at the absolutely continuous points of the boundaries of their discs of representations. The Lyapunov order plays a crucial role in our analysis.

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arXiv2010

Progress in noncommutative function theory

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{*}$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory.

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arXiv2010

Morita Transforms of Tensor Algebras

We show that if $M$ and $N$ are $C^{*}$-algebras and if $E$ (resp. $F$) is a $C^{*}$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$. We show that this functor maps absolutely continuous Hilbert modules to absolutely continuous Hilbert modules and provides a new interpretation of Popescu's reconstruction operator.

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arXiv2010

Representations of Hardy Algebras: Absolute Continuity, Intertwiners and Superharmonic Operators

Suppose $\mathcal{T}_{+}(E)$ is the tensor algebra of a $W^{*}$-correspondence $E$ and $H^{\infty}(E)$ is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of $\mathcal{T}_{+}(E)$ on a Hilbert space to ultra-weakly continuous completely contractive representations of $H^{\infty}(E)$ on the same Hilbert space. Our work extends the classical Sz.-Nagy - Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu's noncommutative disc algebra.

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arXiv2009

Subproduct systems

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 conditions for a semigroup of CP maps to have a *-endomorphic dilation; (iii) an analogue of Parrot's example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no *-endomorphic dilation (thereby solving an open problem raised by Bhat in 1998). Special attention is given to subproduct systems over the semigroup N, which are used as a framework for studying tuples of operators satisfying homogeneous polynomial relations, and the operator algebras they generate. As applications we obtain a noncommutative (projective) Nullstellansatz, a model for tuples of operators subject to homogeneous polynomial relations, a complete description of all representations of Matsumoto's subshift C*-algebra when the subshift is of finite type, and a classification of certain operator algebras -- including an interesting non-selfadjoint generalization of the noncommutative tori.

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arXiv2007

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\mathbb{D}((E^σ)^*)$, of the ``functions'' coming from $H^{\infty}(E)$. We present results that are natural generalizations of the Poisson integral formuala. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the ``model space'' for the canonical model that can be attached to a point in the disc $\mathbb{D}((E^σ)^*)$. We also connect our Poission kernel to various "point evaluations" and to the idea of curvature.

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arXiv2007

Schur Class Operator Functions and Automorphisms of Hardy Algebras

Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ be the associated Hardy algebra. If $σ$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual correspondence $E^σ$ and represent elements in $H^{\infty}(E)$ as $B(H)$-valued functions on the unit ball $\mathbb{D}(E^σ)^{\ast}$. The functions that one obtains are called Schur class functions and may be characterized in terms of certain Pick-like kernels. We study these functions and relate them to system matrices and transfer functions from systems theory. We use the information gained to describe the automorphism group of $H^{\infty}(E)$ in terms of special Möbius transformations on $\mathbb{D}(E^σ)$. Particular attention is devoted to the $H^{\infty}% $-algebras that are associated to graphs.

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arXiv2007

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form \[ L_{e_i}L_{f_j} = \sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}\] where $u= (u_{i,j,k,l})$ is an $nm \times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.

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