We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers.

Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$-algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

For a state $\omega$ on a C$^{*}$-algebra $A$, we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \varphi$, where $\varphi$ is a map on $A$ of the form $\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,$ $\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1$ ($a_i\in A$, $n\in \mathbb {N}$). These are precisely the states $\rho$ that satisfy $\|\rho |J\|\leq \|\omega |J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $\mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$, where $\psi$ is a quantum channel on $\mathcal {R}$ (that is, a map of the form $\psi (x)=\sum \nolimits _ja_j^{*}xa_j$, where $a_j\in \mathcal {R}$ are such that the sum $\sum \nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^{*}$-algebra and for properly infinite von Neumann algebras the converse also holds.

Given a self-similar set K defined from an iterated function system $\Gamma =(\gamma _{1},\ldots ,\gamma _{d})$ and a set of functions $H=\{h_{i}:K\to \mathbb {R}\}_{i=1}^{d}$ satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras $\mathcal {O}_{\Gamma }$ and their Toeplitz extensions $\mathcal {T}_{\Gamma }$ . We then characterize the KMS states for this action. For each $\beta \in (0,\infty )$ , there is a Ruelle operator $\mathcal {L}_{H,\beta }$ , and the existence of KMS states at inverse temperature $\beta $ is related to this operator. The critical inverse temperature $\beta _{c}$ is such that $\mathcal {L}_{H,\beta _{c}}$ has spectral radius 1. If $\beta <\beta _{c}$ , there are no KMS states on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ ; if $\beta =\beta _{c}$ , there is a unique KMS state on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ which is given by the eigenmeasure of $\mathcal {L}_{H,\beta _{c}}$ ; and if $\beta>\beta _{c}$ , including $\beta =\infty $ , the extreme points of the set of KMS states on $\mathcal {T}_{\Gamma }$ are parametrized by the elements of K and on $\mathcal {O}_{\Gamma }$ by the set of branched points.

We consider symmetry-protected topological phases with on-site finite group G symmetry $\beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$-valued invariant.

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

We study the invariance of KMS states on graph $C^{\ast }$-algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$-algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C*-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.

We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in $\mathbb {Z}_2 \times H^1(G,\mathbb {Z}_2) \times H^2(G, U(1)_{\mathfrak {p}})$ with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.

Let $\phi :X\to X$ be a homeomorphism of a compact metric space X. For any continuous function $F:X\to \mathbb {R}$ there is a one-parameter group $\alpha ^{F}$ of automorphisms (or a flow) on the crossed product $C^*$ -algebra $C(X)\rtimes _{\phi }\mathbb {Z}$ defined such that $\alpha ^{F}_{t}(fU)=fUe^{-itF}$ when $f \in C(X)$ and U is the canonical unitary in the construction of the crossed product. In this paper we study the Kubo--Martin--Schwinger (KMS) states for these flows by developing an intimate relation to the ergodic theory of non-singular transformations and show that the structure of KMS states can be very rich and complicated. Our results are complete concerning the set of possible inverse temperatures; in particular, we show that when $C(X) \rtimes _{\phi } \mathbb Z$ is simple this set is either $\{0\}$ or the whole line $\mathbb R$ .

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

The modular Gromov–Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov–Hausdorff propinquity.

We study restriction and extension properties for states on ${{\text{C}}^{*}}$-algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson’s hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a ${{\text{C}}^{*}}$-algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient ${{\text{C}}^{*}}$-algebra. We prove that commuting pairs are unperforated and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.

The notion of a strongly dense inner product space is introduced and it is shown that, for such an incomplete space $S$ (in particular, for each incomplete hyperplane of a Hilbert space), the system $F(S)$ of all orthogonally closed subspaces of $S$ is not stateless, and the state-space of $F(S)$ is affinely homeomorphic to the face consisting of the free states on the projection lattice corresponding to the completion of $S$. The homeomorphism is determined by the extension of the states. In particular, when $S$ is complex, the state-space of $F(S)$ is affinely homeomorphic to the state-space of the Calkin algebra associated with $\skew3\overline S$.

Given a sequence $(\pi_n)$ of irreducible representations of a liminal $C^*$-algebra $A$, and a sequence $(b_n)$ of trace class operators with $b_n\in \pi_n(A)$, we investigate necessary conditions and sufficient conditions for the existence of a simultaneous lifting $a\in A$ such that, for each $n$, the trace of $\sigma(a)$ is bounded for irreducible representations $\sigma$ in a neighbourhood of $\pi_n$.

We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple ${{C}^{*}}$-algebras. The first result of this kind was shown by Powers for the $\text{UHF}$ algbras some 30 years ago.

We investigate representations of the Cuntz algebra ${{\mathcal{O}}_{2}}$ on antisymmetric Fock space ${{F}_{a}}({{\mathcal{K}}_{1}})$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi {{P}_{1}}$. We pay special attention to the vector states on ${{\mathcal{O}}_{2}}$ corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra ${{\mathcal{F}}_{2}}$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi {{P}_{1}}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to ${{\mathcal{F}}_{2}}$.

The endomorphisms of $B\left( {{F}_{a}}({{\mathcal{K}}_{1}}) \right)$ associated with these representations of ${{\mathcal{O}}_{2}}$ are also considered.