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Evgenios T. A. Kakariadis

Publications and source records attributed to Evgenios T. A. Kakariadis.

At least 37 records · Page 2Linked to original sources
arXiv2019

Entropy theory for the parametrization of the equilibrium states of Pimsner algebras

We consider Pimsner algebras that arise from C*-correspondences of finite rank, as dynamical systems with their rotational action. We revisit the Laca-Neshveyev classification of their equilibrium states at positive inverse temperature along with the parametrizations of the finite and the infinite parts simplices by tracial states on the diagonal. The finite rank entails an entropy theory that shapes the KMS-structure. We prove that the infimum of the tracial entropies dictates the critical inverse temperature, below which there are no equilibrium states for all Pimsner algebras. We view the latter as the entropy of the ambient C*-correspondence. This may differ from what we call strong entropy, above which there are no equilibrium states of infinite type. In particular, when the diagonal is abelian then the strong entropy is a maximum critical temperature for those. In this sense we complete the parametrization method of Laca-Raeburn and unify a number of examples in the literature.

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arXiv2018

On operator algebras associated with monomial ideals in noncommuting variables

We study operator algebras arising from monomial ideals in the ring of polynomials in noncommuting variables, through the apparatus of subproduct systems and C*-correspondences. We provide a full comparison amongst the related operator algebras. For our analysis we isolate a partially defined dynamical system, to which we refer as the {\em quantised dynamics} of the monomial ideal. In addition we revisit several previously considered constructions. These include Matsumoto's subshift C*-algebras, as well as the tensor and the Pimsner algebras associated with dynamical systems or graphs. We sort out the various relations by giving concrete conditions and counterexamples that orientate the operator algebras of our context. It appears that the boundary C*-algebras do not arise as the quotient with the compact operators unconditionally. We establish a dichotomy to this effect by examining the resulting tensor algebras. We identify their boundary representations, we analyse their C*-envelopes, and we give criteria for hyperrigidity. Moreover we completely classify them in terms of the data provided by the monomial ideals. For tensor algebras of C*-correspondences and bounded isomorphisms this is achieved up to the level of local conjugacy (in the sense of Davidson and Roydor) for the quantised dynamics. For tensor algebras of subproduct systems and algebraic isomorphisms this is achieved up to the level of equality of monomial ideals modulo permutations of the variables. In the process we accomplish more in different directions. Most notably we show that tensor algebras form a complete invariant for isomorphic (resp. similar) subproduct systems of homogeneous ideals up to isometric (resp. bounded) isomorphisms. The results on local conjugacy are obtained via an alternative proof of the breakthrough result of Davidson and Katsoulis on piecewise conjugate systems. For our purposes we use appropriate compressions of the Fock representation. We then apply this alternative proof locally for the partially defined quantised dynamics. In this way we avoid the topological graphs machinery and pave the way for further applications. These include operator algebras of dynamical systems over commuting contractions or over row commuting contractions.

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arXiv2017

A proof of Boca's Theorem

We give a general method of extending unital completely positive maps to amalgamated free products of C*-algebras. As an application we give a dilation theoretic proof of Boca's Theorem.

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arXiv2017

Free multivariate w*-semicrossed products: reflexivity and the bicommutant property

We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products over factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if so does the ambient algebra of the dynamics.

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arXiv2017

On the quantized dynamics of factorial languages

We study local piecewise conjugacy of the quantized dynamics arising from factorial languages. We show that it induces a bijection between allowable words of same length and thus it preserves entropy. In the case of sofic factorial languages we prove that local piecewise conjugacy translates to unlabeled graph isomorphism of the follower set graphs. Moreover it induces an unlabeled graph isomorphism between the Fischer covers of irreducible subshifts. We verify that local piecewise conjugacy does not preserve finite type nor irreducibility; but it preserves soficity. Moreover it implies identification (up to a permutation) for factorial languages of type $1$ if, and only if, the follower set function is one-to-one on the symbol set.

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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

Strong Morita equivalence of operator spaces

We introduce and examine the notions of strong $Δ$-equivalence and strong TRO equivalence for operator spaces. We show that they behave in an analogous way to how strong Morita equivalence does for the category of C*-algebras. In particular, we prove that strong $Δ$-equivalence coincides with stable isomorphism under the expected countability hypothesis, and that strongly TRO equivalent operator spaces admit a correspondence between particular representations. Furthermore we show that strongly $Δ$-equivalent operator spaces have stably isomorphic second duals and strongly $Δ$-equivalent TRO envelopes. In the case of unital operator spaces, strong $Δ$-equivalence implies stable isomorphism of the C*-envelopes.

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arXiv2016

On Nica-Pimsner algebras of C*-dynamical systems over $\mathbb{Z}_+^n$

We examine Nica-Pimsner algebras associated with semigroup actions of $\mathbb{Z}_+^n$ on a C*-algebra $A$ by $*$-endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore we parameterize the KMS states at finite temperature by the tracial states on $A$. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on $A$ (resp. states on $A$). Finally we give a formula for obtaining tracial states on the Nica-Pimsner algebras.

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arXiv2016

Semicrossed Products of Operator Algebras by Semigroups

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope for various semicrossed products of (possibly nonselfadjoint) operator algebras by spanning cones and lattice-ordered abelian semigroups. In particular, we show that the C*-envelope of the semicrossed product of C*-dynamical systems by doubly commuting representations of $\mathbb{Z}^n_+$ (by generally non-injective endomorphisms) is the full corner of a C*-crossed product. In consequence we connect the ideal structure of C*-covers to properties of the actions. In particular, when the system is classical, we show that the C*-envelope is simple if and only if the action is injective and minimal. The dilation methods that we use may be applied to non-abelian semigroups. We identify the C*-envelope for actions of the free semigroup $\mathbb{F}_+^n$ by automorphisms in a concrete way, and for injective systems in a more abstract manner. We also deal with C*-dynamical systems over Ore semigroups when the appropriate covariance relation is considered.

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arXiv2015

KMS states on Pimsner algebras associated with C*-dynamical systems

We examine the theory of the KMS states on Pimsner algebras arising from multivariable unital C*-dynamical systems. As an application we show that Pimsner algebras of piecewise conjugate classical systems attain the same KMS states, even though it is still an open problem whether or not they are $*$-isomorphic.

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arXiv2014

Ergodic extensions and Hilbert modules associated to endomorphisms of MASAS

We show that a class of ergodic transformations on a probability measure space $(X,μ)$ extends to a representation of $\mathcal{B}(L^2(X,μ))$ that is both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in this work. During the analysis of the existence and uniqueness of such a Cuntz family we give several results of individual interest. Most notably we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal basis of Hilbert modules. We remark that the trivial Hilbert module of the Cuntz algebra $\mathcal{O}_N$ does not have a well-defined Hilbert module basis (moreover that it is unitarily equivalent to the module sum $\sum_{i=1}^n \mathcal{O}_N$ for infinitely many $n \in \mathbb{N}$).

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arXiv2014

Semicrossed products of C*-algebras and their C*-envelopes

Let $\mathcal{C}$ be a C*-algebra and $α:\mathcal{C} \rightarrow \mathcal{C}$ a unital *-endomorphism. There is a natural way to construct operator algebras which are called semicrossed products, using a convolution induced by the action of $α$ on $\mathcal{C}$. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that, when $α$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product $\mathcal{C}_\infty \rtimes_{α_\infty} \mathbb{Z}$. We show that minimality of the dynamical system $(\mathcal{C},α)$ is equivalent to non-existence of non-trivial Fourier invariant ideals in the C*-envelope. We get sharper results for commutative dynamical systems.

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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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arXiv2014

Representations of C*-dynamical systems implemented by Cuntz families

Given a dynamical system $(A,\al)$ where $A$ is a unital $\ca$-algebra and $\al$ is a (possibly non-unital) *-endomorphism of $A$, we examine families $(π,\{T_i\})$ such that $π$ is a representation of $A$, $\{T_i\}$ is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of non-selfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest $\ca$-algebra they generate, namely the $\ca$-envelope. We then relate each occurrence of the $\ca$-envelope to (a full corner of) an appropriate twisted crossed product. We provide a counterexample to show the extent of this variety. In the context of $\ca$-algebras, these results can be interpreted as analogues of Stacey's famous result, for non-automorphic systems and $n>1$. Our study involves also the one variable generalized crossed products of Stacey and Exel. In particular, we refine a result that appears in the pioneering paper of Exel on (what is now known as) Exel systems.

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arXiv2014

Semicrossed Products of Operator Algebras: A Survey

Semicrossed product algebras have been used to study dynamical systems since their introduction by Arveson in 1967. In this survey article, we discuss the history and some recent work, focussing on the conjugacy problem, dilation theory and C*-envelopes, and some connections back to the dynamics

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arXiv2013

The Šilov Boundary for Operator Spaces

Motivated by the recent interest in the examination of unital completely positive maps and their effects in C*-theory, we revisit an older result concerning the existence of the Šilov ideal. The direct proof of Hamana's theorem for the existence of an injective envelope for a unital operator subspace X of some B(H) that we provide implies that the Šilov ideal is the intersection of C*(X) with any maximal boundary operator subsystem in B(H). As an immediate consequence we deduce that the Šilov ideal is the biggest boundary operator subsystem for X in C*(X). The new proof of the existence of the Šilov ideal that we give does not use the existence of maximal dilations, provided by Dritschel and McCullough, and so it is independent of the one given by Arveson. As a countereffect, the Šilov ideal can be seen as the set that contains the abnormalities in a C*-cover (C,ι) of X for all the extensions of the identity map on ι(X). The interpretation of our results in terms of ucp maps characterizes the maximal boundary subsystems of X in B(H) as kernels of X-projections that induce completely minimal X-seminorms; equivalently, X-minimal projections with range being an injective envelope, that we view from now on as the Šilov boundary for X.

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arXiv2013

The Dirichlet Property for Tensor Algebras

We prove that the tensor algebra of a C*-correspondence $X$ is Dirichlet if and only if $X$ is a Hilbert bimodule. As a consequence, we point out and fix an error appearing in the proof of a famous result of Duncan. Secondly we answer a question raised by Davidson and Katsoulis concerning tensor algebras and semi-Dirichlet algebras, by giving an example of a Dirichlet algebra that cannot be described as the tensor algebra of any C*-correspondence. Furthermore we show that the adding tail technique, as extended by the author and Katsoulis, applies in a unique way to preserve the class of Hilbert bimodules. The exploitation of these ideas implies that the tensor algebra of row-finite graphs, the tensor algebra of multivariable automorphic $\ca$- dynamics and Peters' semicrossed product of an injective $\ca$-dynamical system have the unique extension property. The two latter provide examples of non-separable operator algebras that admit a Choquet boundary in the sense of Arveson.

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