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Elias G. Katsoulis

Publications and source records attributed to Elias G. Katsoulis.

At least 19 recordsLinked to original sources
arXiv2026

Rigidity for Isomorphisms between Operator Algebras with Commutative Diagonals

We show that two families of operator algebras, the CSL algebras of multiplicity free CSLs and the semicrossed products of commutative C$^*$-algebras, demonstrate a strong form of rigidity with respect to isometric isomorphisms. Specifically, the isomorphism class of any such algebra remains unchanged within its family, even if we allow for isomorphism after tensoring with operator algebras containing the compact operators. For semicrossed products of commutative C$^*$-algebras, the same conclusion holds even when tensoring with operator algebras whose diagonals are irreducibly acting. Collectively, these results imply rigidity with respect to stable isomorphisms: two algebras are isometrically isomorphic if and only if they are stably isomorphic.

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arXiv2025

Crossed products and C*-covers of semi-Dirichlet operator algebras

In this paper, we show that the semi-Dirichlet C*-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C*-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.

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arXiv2024

On Toeplitz algebras of product systems

In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right LCM monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear iff the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens iff the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag-Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for $m$ and $n$ relatively prime.

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arXiv2022

C*-algebras and Equivalences for C*-correspondences

We study several notions of shift equivalence for C*-correspondences and the effect that these equivalences have on the corresponding Pimsner dilations. Among others, we prove that non-degenerate, regular, full C*-correspondences which are shift equivalent have strong Morita equivalent Pimsner dilations. We also establish that the converse may not be true. These results settle open problems in the literature. In the context of C*-algebras, we prove that if two non-degenerate, regular, full C*-correspondences are shift equivalent, then their corresponding Cuntz-Pimsner algebras are strong Morita equivalent. This generalizes results of Cuntz and Krieger and Muhly, Tomforde and Pask. As a consequence, if two subshifts of finite type are eventually conjugate, then their Cuntz-Krieger algebras are strong Morita equivalent. Our results suggest a natural analogue of the Shift Equivalence Problem in the context of C*-correspondences. Even though we do not resolve the general Shift Equivalence Problem, we obtain a positive answer for the class of imprimitivity bimodules.

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arXiv2022

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

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arXiv2021

Co-universality and controlled maps on product systems over right LCM-semigroups

We study the structure of C*-algebras associated with compactly aligned product systems over group embeddable right LCM-semigroups. Towards this end we employ controlled maps and a controlled elimination method that associates the original cores to those of the controlling pair, and we combine with applications of the C*-envelope theory for cosystems of nonselfadjoint operator algebras recently produced. We derive several applications of these methods that generalize results on single C*-correspondences. First we show that if the controlling group is exact then the co-universal C*-algebra of the product system coincides with the quotient of the Fock C*-algebra by the ideal of strong covariance relations. We show that if the controlling group is amenable then the product system is amenable. In particular if the controlling group is abelian then the co-universal C*-algebra is the C*-envelope of the tensor algebra. Secondly we give necessary and sufficient conditions for the Fock C*-algebra to be nuclear and exact. When the controlling group is amenable we completely characterize nuclearity and exactness of any equivariant injective Nica-covariant representation of the product system. Thirdly we consider controlled maps that enjoy a saturation property. In this case we induce a compactly aligned product system over the controlling pair that shares the same Fock representation, and preserves injectivity. By using co-universality, we show that they share the same reduced covariance algebras. If in addition the controlling pair is a total order then the fixed point algebra of the controlling group induces a super-product system that has the same reduced covariance algebra and is moreover reversible.

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arXiv2021

Boundary quotient C*-algebras of semigroups

We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of $P$ that reflect the space $\mathcal{J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation $\mathrm{C}^*_λ(P)$ and we consider the boundary quotient $\partial \mathrm{C}^*_λ(P)$ related to the minimal boundary space. We show that $\partial \mathrm{C}^*_λ(P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of $P$; and (b) with respect to the equivariant C*-covers of the reduced nonselfadjoint semigroup algebra $\mathcal{A}(P)$. If $P$ is an Ore semigroup, or if $G$ acts topologically freely on the minimal boundary space, then $\partial \mathrm{C}^*_λ(P)$ coincides with the usual C*-envelope $\mathrm{C}^*_{\text{env}}(\mathcal{A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of nonzero algebraic integers, and the $ax+b$-semigroups over integral domains that are not a field. In particular, we show that $P$ is an Ore semigroup if and only if there exists a canonical $*$-isomorphism from $\partial \mathrm{C}^*_λ(P)$, or from $\mathrm{C}^*_{\text{env}}(\mathcal{A}(P))$, onto $\mathrm{C}^*_λ(G)$. If any of the above holds, then $\mathcal{A}(P)$ is shown to be hyperrigid.

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arXiv2020

Operator algebras of higher rank numerical semigroups

A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem to the negative. Here we show that these algebras share the polydisc as the character space in a canonical way. We subsequently use this feature in order to identify higher rank numerical semigroups from the corresponding nonselfadjoint algebras.

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arXiv2017

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong $Δ$-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a $σ$-TRO. As an application we show that strong Morita equivalence coincides with strong $Δ$-equivalence for tensor algebras of aperiodic C*-correspondences.

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arXiv2016

C*-envelopes and the Hao-Ng isomorphism for discrete groups

Using non-selfadjoint techniques, we establish the Hao-Ng isomorphism for the reduced crossed product and all discrete groups. For the full crossed product an analogous result holds for all discrete groups but the C*-correspondences involved have to be hyperrigid. These results are obtained by calculating the C*-envelope of the reduced crossed product of an operator algebra by a discrete group.

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arXiv2015

Local maps and the representation theory of operator algebras

Using representation theory techniques we prove that various spaces of derivations or one-sided multipliers over certain operator algebras are reflexive. A sample result: any bounded local derivation (local left multiplier) on an automorphic semicrossed product is a derivation (resp. left multiplier). In the process we obtain various results of independent interest. In particular, the finite dimensional nest representations of the tensor algebra of a topological graph separate points.

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arXiv2015

The reflexive closure of the adjointable operators

Given a Hilbert module E over a C*-algebra A, we show that the collection of all bounded A-module operators acting on E forms the reflexive closure for the algebra of the adjointable operators. We also make an observation regarding the representation theory of the left centralizer algebra of a C*-algebra and use it to give an intuitive proof of a related result of H. Lin

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arXiv2014

Isomorphism Invariants for Multivariable C*-Dynamics

To a given multivariable C*-dynamical system $(A, \al)$ consisting of *-automorphisms, we associate a family of operator algebras $\alg(A, \al)$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A}, \hat{\al})$ and $(\hat{B}, \hat{\be})$ on the Fell spectra are piecewise conjugate, in the sense of Davidson and Katsoulis. In the course of proving the above theorem we obtain several results of independent interest. If $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the associated correspondences $X_{(A, \al)}$ and $X_{(B, \be)}$ are unitarily equivalent. In particular, the tensor algebras are isometrically isomorphic if and only if the associated correspondences are unitarily equivalent. Furthermore, isomorphism of semicrossed products implies isomorphism of the associated tensor algebras. In the case of multivariable systems acting on C*-algebras with trivial center, unitary equivalence of the associated correspondences reduces to outer conjugacy of the systems. This provides a complete invariant for isometric isomorphisms between semicrossed products as well.

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arXiv2011

Dilation theory, commutant lifting and semicrossed products

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. This is applied to the study of semicrossed products by automorphisms, and endomorphisms which lift to the C*-envelope. In particular, we obtain several general theorems which allow one to conclude that semicrossed products of an operator algebra naturally imbed completely isometrically into the semicrossed product of its C*-envelope, and the C*-envelopes of these two algebras are the same.

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arXiv2011

Semicrossed products of the disc algebra

If $α$ is the endomorphism of the disk algebra, $\AD$, induced by composition with a finite Blaschke product $b$, then the semicrossed product $\AD\times_α \bZ^+$ imbeds canonically, completely isometrically into $\rC(\bT)\times_α \bZ^+$. Hence in the case of a non-constant Blaschke product $b$, the C*-envelope has the form $ \rC(§_{b})\times_{s} \bZ$, where $(§_{b}, s)$ is the solenoid system for $(\bT, b)$. In the case where $b$ is a constant, then the C*-envelope of $\AD\times_α \bZ^+$ is strongly Morita equivalent to a crossed product of the form $ \rC(§_{e})\times_{s} \bZ$, where $e \colon \bT \times \bN \longrightarrow \bT \times \bN$ is a suitable map and $(§_{e}, s)$ is the solenoid system for $(\bT \times \bN, \, e)$ .

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arXiv2011

Contributions to the theory of C*-correspondences with applications to multivariable dynamics

Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective C*-correspondence and the dilation of an injective C*-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C*-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C*-dynamics. We also exhibit a transparent picture for the dilation of an injective C*-correspondence to an essential Hilbert bimodule. As an application of our constructs, we prove two results in the theory of multivariable dynamics that extend results of Davidson and Roydor and Peters. We also discuss the impact of our results on the description of the C*-envelope of a tensor algebra as the Cuntz-Pimsner algebra of the associated C*-correspondence.

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