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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.

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.

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., Comment: 40 pages

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 $\mathrm{C}^*_\lambda(P)$ and we consider the boundary quotient $\partial \mathrm{C}^*_\lambda(P)$ related to the minimal boundary space. We show that $\partial \mathrm{C}^*_\lambda(P)$ 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 $\partial \mathrm{C}^*_\lambda(P)$ 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 $\partial \mathrm{C}^*_\lambda(P)$, or from $\mathrm{C}^*_{\text{env}}(\mathcal{A}(P))$, onto $\mathrm{C}^*_\lambda(G)$. If any of the above holds, then $\mathcal{A}(P)$ is shown to be hyperrigid., Comment: 22 pages. Theorem 4.5 to $ax+b$-semigroups of rings has a more general application than originally stated, leading to the updated Remark 4.8(vi)

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., Comment: 25 pages, accepted version, Advances in Mathematics, to appear

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., Comment: 10 pages, minor corrections

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 $\Delta$-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 $\sigma$-TRO. As an application we show that strong Morita equivalence coincides with strong $\Delta$-equivalence for tensor algebras of aperiodic C*-correspondences., Comment: 20 pages, minor corrections, additional comments in subsection 2.4

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., Comment: 15 pages, final version to appear in International Mathematics Research Notices

This paper is an expanded version of the lectures I delivered at the Indian Statistical Institute, Bangalore, during the OTOA 2014 conference., Comment: Expository article, 56 pages, comments welcome. Many clarifications and corrections in Section 5

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., Comment: The paper will appear in the Transactions of the AMS. Several typos corrected. 23 pages

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, Comment: The paper will appear in the Illinois J. Math

In this paper we survey our recent work on C*-correspondences and their associated operator algebras; in particular, on adding tails, the Shift Equivalence Problem and Hilbert bimodules., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1101.1482

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., Comment: 3+33pp. We add a corrigendum to the original entry, in compliance to the arXiv policy, concerning a gap in the proof of Theorem 5.8, that results to a weaker version of SE than the one sought. In the corrigendum we show where the problem lies, adjust the main results --which still tackle the two conjectures of the introduction in the original entry-- and make connections with current literature

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., Comment: 16 pages; changes in the Introduction and in Theorem 5.2

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.

If $\alpha$ is the endomorphism of the disk algebra, $\AD$, induced by composition with a finite Blaschke product $b$, then the semicrossed product $\AD\times_{\alpha} \bZ^+$ imbeds canonically, completely isometrically into $\rC(\bT)\times_{\alpha} \bZ^+$. Hence in the case of a non-constant Blaschke product $b$, the C*-envelope has the form $ \rC(\S_{b})\times_{s} \bZ$, where $(\S_{b}, s)$ is the solenoid system for $(\bT, b)$. In the case where $b$ is a constant, then the C*-envelope of $\AD\times_{\alpha} \bZ^+$ is strongly Morita equivalent to a crossed product of the form $ \rC(\S_{e})\times_{s} \bZ$, where $e \colon \bT \times \bN \longrightarrow \bT \times \bN$ is a suitable map and $(\S_{e}, s)$ is the solenoid system for $(\bT \times \bN, \, e)$ ., Comment: 7 pages

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., Comment: 32 pages

Let $(\A, \alpha)$ and $(\B, \beta)$ be C*-dynamical systems and assume that $\A$ is a separable simple C*-algebra and that $\alpha$ and $\beta$ are *-automorphisms. Then the semicrossed products $\A \times_{\alpha} \bbZ^{+}$ and $\B \times_{\beta} \bbZ^{+}$ are isometrically isomorphic if and only if the dynamical systems $(\A, \alpha)$ and $(\B, \beta)$ are outer conjugate., Comment: 12 pages, accepted for publication in Math. Ann

Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\tau_i:X \to X$ for $1 \le i \le n$. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\A(X, \tau)$ and the semicrossed product $\rC_0(X)\times_\tau\Fn$. We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from either the algebraic structure of $\A(X, \tau)$ or $\rC_0(X)\times_\tau\Fn$. Various classification results follow as a consequence. For example, for $n=2,3$, the tensor algebras are (algebraically or even completely isometrically) isomorphic if and only if the systems are piecewise topologically conjugate. In order to establish these results we make use of analytic varieties as well as homotopy theory for Lie groups We define a generalized notion of wandering sets and recurrence. Using this, it is shown that $\A(X, \tau)$ or $\rC_0(X)\times_\tau\Fn$ is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each $\tau_i$ being surjective and $v$-recurrent points being dense for each $v \in \Fn$., Comment: In this version we explicitly exhibit enough (irreducible) boundary repns to faithfully represent the C*-envelope of the tensor algebra of the system. We also show that piecewise conjugacy is not a complete invariant for complete isomorphisms between semicrossed products. The paper has been reorganized

A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that $\eta_i:\X_i\to \X_i$ is a continuous proper map on a locally compact Hausdorff space $\X_i$, for $i = 1,2$. We show that the dynamical systems $(\X_1, \eta_1)$ and $(\X_2, \eta_2)$ are conjugate if and only if some topological conjugacy algebra of $(\X_1, \eta_1)$ is isomorphic as an algebra to some topological conjugacy algebra of $(\X_2, \eta_2)$. This implies as a corollary the complete classification of the semicrossed products $C_0(\X) \times_{\eta} \bbZ^{+}$, which was previously considered by Arveson and Josephson, Peters, Hadwin and Hoover and Power. We also obtain a complete classification of all semicrossed products of the form $A(\bbD) \times_{\eta}\bbZ^{+}$, where $A(\bbD)$ denotes the disc algebra and $\eta: \bbD \to \bbD$ a continuous map which is analytic on the interior. In this case, a surprising dichotomy appears in the classification scheme, which depends on the fixed point set of $\eta$. We also classify more general semicrossed products of uniform algebras., Comment: 25 pages. Accepted for publication in Crelle's Journal

We show that the $\ca$-envelope of the tensor algebra of an arbitrary $\ca$-correspondence $\X$ coincides with the Cuntz-Pimsner algebra $\O_{\X}$, as defined by Katsura \cite{Ka}. This improves earlier results of Muhly and Solel and Fowler, Muhly and Raeburn, who came to the same conclusion under the additional hypothesis that $\X$ is strict and faithful., Comment: 9 pages, final version, Journal of Functional Analysis, to appear

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider consequences of this result. In particular, we give partial solutions to a conjecture of Power [Limit Algebras, Longman, 1992, Notes to Chapter 8] and to an open problem [op. cit., Problem 7.8]. As a further consequence, we describe epimorphisms between various classes of limit algebras., Comment: 13 pages