Equilibrium states and entropy theory for Nica-Pimsner algebras

We study the equilibrium simplex of Nica-Pimsner algebras arising from product systems of finite rank on the free abelian semigroup. First we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube. Secondly we associate every gauge-invariant part to a sub-simplex of tracial states of the diagonal algebra. We show how this parametrization lifts to the full equilibrium simplices of non-infinite type. The finite rank entails an entropy theory for identifying the two critical inverse temperatures: (a) the least upper bound for existence of non finite-type equilibrium states, and (b) the least positive inverse temperature below which there are no equilibrium states at all. We show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies (modulo negative values). Thus phase transitions can happen only in-between these two critical points and possibly at zero temperature.Comment: 42 pages; corrected typo

Finite dimensional approximations for Nica-Pimsner algebras

We give necessary and sufficient conditions for nuclearity of Cuntz-Nica-Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag-Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica-Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the co-efficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz-Nica-Pimsner algebra and the Cuntz-Nica-Pimsner algebra. We complete this study with the relevant results on exactness.Comment: 22 page

A Note on the Gauge Invariant Uniqueness Theorem for C*-correspondences

We present a short proof of the gauge invariant uniqueness theorem for relative Cuntz-Pimsner algebras of C*-correspondences.Comment: 7 pages; changes in the introductio

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

Let $\mathcal{C}$ be a C*-algebra and $\alpha:\mathcal{C} \rightarrow \mathcal{C}$ 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 $\alpha$ 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 $\alpha$ is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product $\mathcal{C}_\infty \rtimes_{\alpha_\infty} \mathbb{Z}$. We show that minimality of the dynamical system $(\mathcal{C},\alpha)$ is equivalent to non-existence of non-trivial Fourier invariant ideals in the C*-envelope. We get sharper results for commutative dynamical systems.Comment: 27 pages, changes around theorem 4.

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.Comment: 40 pages; added section about how the entropy theory applies to reducible graphs to obtain the KMS-simplices of an Huef-Laca-Raeburn-Sim

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.Comment: 9 pages, minor corrections in tex

Ergodic extensions and Hilbert modules associated to endomorphisms of MASAS

We show that a class of ergodic transformations on a probability measure space $(X,\mu)$ extends to a representation of $\mathcal{B}(L^2(X,\mu))$ 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 $n \in \mathbb{N}$).Comment: 14 page

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.Comment: 77 pages, changes in the format, added Proposition 10.2 that connects the Cuntz-Pimsner algebra of a monomial ideal with the Cuntz-Krieger algebra of the unlabeled graph of the follower set graph of the idea

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.Comment: 32 page

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.Comment: 16 pages; changes in the Introduction and in Theorem 5.