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Register a new userFinite dimensional approximations for Nica-Pimsner algebras
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Authors
Evgenios T.A. Kakariadis
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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.
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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.
The residual finite-dimensionality of a $mathrm{C}^*$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of whether residual finite-dimensionality of an operator algebra is inherited by its maximal $mathrm{C}^*$-cover, which we resolve in many cases of interest.
We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a new proof that the K-theory of a 2-graph C*-algebra is independent of the factorisation rules, and we further show that the K-theory of any twisted k-graph C*-algebra is independent of the twisting 2-cocycle. We also explore applications to K-theory for the C*-algebras of single-vertex k-graphs, reducing the question of whether the $K$-theory is independent of the factorisation rules to a question about path-connectedness of the space of solutions to an equation of Yang-Baxter type.
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.
In this paper we study weight
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