KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

Given a quasi-lattice ordered group (G, P) and a compactly aligned product system X of essential $$\hbox {C}^*$$ C ∗ -correspondences over the monoid P, we show that there is a bijection between the gauge-invariant $$\hbox {KMS}_\beta $$ KMS β -states on the Nica-Toeplitz algebra $$\mathcal {NT}(X)$$ NT ( X ) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature $$\beta $$ β . Under fairly general additional assumptions we show that there is a critical inverse temperature $$\beta _c$$ β c such that for $$\beta >\beta _c$$ β > β c all $$\hbox {KMS}_\beta $$ KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of $$\hbox {KMS}_\beta $$ KMS β -states in terms of tracial states on A, while at $$\beta =\beta _c$$ β = β c we have a phase transition manifesting itself in the appearance of $$\hbox {KMS}_\beta $$ KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on $$\mathcal {NT}(X)$$ NT ( X ) .

Factorizable Maps and Traces on the Universal Free Product of Matrix Algebras

We relate factorizable quantum channels on $M_n({\mathbb{C}})$, for $n \ge 2$, via their Choi matrix, to certain matrices of correlations, which, in turn, are shown to be parametrized by traces on the unital free product $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$. Factorizable maps with a finite dimensional ancilla are parametrized by finite dimensional traces on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$, and factorizable maps that approximately factor through finite dimensional $C^\ast $-algebras are parametrized by traces in the closure of the finite dimensional ones. The latter set of traces is shown to be equal to the set of hyperlinear traces on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$. We finally show that each metrizable Choquet simplex is a face of the simplex of tracial states on $M_n({\mathbb{C}}) \ast _{\mathbb{C}} M_n({\mathbb{C}})$.

Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

Traces on topological-graph algebras

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ and Radon probability measures on the vertex space $E^{0}$ which are, in a suitable sense, invariant under the action of the edge space $E^{1}$. It is shown that if $E$ has no cycles, then every tracial state on $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.

TRACES ARISING FROM REGULAR INCLUSIONS

We study the problem of extending a state on an abelian $C^{\ast }$-subalgebra to a tracial state on the ambient $C^{\ast }$-algebra. We propose an approach that is well suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^{\ast }$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^{\ast }$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz–Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provide connections to related finiteness questions in graph $C^{\ast }$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

Supramenable groups and partial actions

We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a $\text{C}^{\ast }$-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.