arXiv subjects
Publications and source records attributed to Chris Bruce.
For every Cuntz--Krieger groupoid, we show that there is a topologically free boundary action of the outer automorphism group of its topological full group on the Hilbert cube. In particular, these outer automorphism groups, including the outer automorphism groups of all Higman--Thompson groups, are C*-simple.
We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of algebraic integers, and we use our groupoid homology calculation to compute the K-theory for ring C*-algebras of rings of algebraic integers, recovering the results of Cuntz and Li and of Li and L\"uck without using Cuntz-Li duality. Moreover, we compute the K-theory for C*-algebras attached to integral dynamics, resolving the conjecture by Barlak, Omland, and Stammeier in full generality.
We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. \textbf{44} (2022), no. 4, 88--112].
Given a conditional expectation $P$ from a C*-algebra $B$ onto a C*-subalgebra $A$, we observe that induction of ideals via $P$, together with a map which we call co-induction, forms a Galois connection between the lattices of ideals of $A$ and $B$. Using properties of this Galois connection, we show that, given a discrete group $G$ and a stabilizer subgroup $G_x$ for the action of $G$ on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of $C^*(G_x)$ and the set of maximal ideals of $C^*_r(G)$. As an application, we prove that the reduced C*-algebra of Thompson's group $T$ has a unique maximal ideal. Furthermore, we show that, if Thompson's group $F$ is amenable, then $C^*_r(T)$ has infinitely many ideals.
We show that a normal coaction of a discrete group on an operator algebra extends to a normal coaction on the C*-envelope. This resolves an open problem attempted by several experts in the area, and provides a more direct proof of a prominent result of Sehnem. As an application, we resolve a question of X. Li, where we identify the C*-envelopes of the operator algebras of groupoid-embeddable categories and of cancellative right LCM monoids. This latter class includes many examples of monoids that are not group-embeddable.
We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers $p$ and $q$, we show that Furstenberg's $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2)$.
We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions rationally embed in each other. For specific example classes arising for instance from toral endomorphisms, actions from rings, or actions from commutative algebra, this mutual embedability can be improved in various ways to obtain surprisingly strong rigidity phenomena. This is witnessed in a particularly striking fashion for actions arising from algebraic number theory: We prove that the groupoids associated with the action of the multiplicative monoid of non-zero elements in a ring of algebraic integers on the additive group of the ring remembers the initial algebraic action up to isomorphism, which in turn remembers the isomorphism class of the ring. This resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch--Uchida theorem using topological full groups.
The multiplicative group of a number field acts by multiplication on the full adele ring of the field. Generalising a theorem of Laca and Raeburn, we explicitly describe the primitive ideal space of the crossed product C*-algebra associated with this action. We then distinguish real, complex, and finite places of the number field using K-theoretic invariants. Combining these results with a recent rigidity theorem of the authors implies that any *-isomorphism between two such C*-algebras gives rise to an isomorphism of the underlying number fields that is constructed from the *-isomorphism.
We provide a framework for studying concrete C*-algebras associated with algebraic actions of semigroups: Given such an action, we construct an inverse semigroup, and we introduce conditions for algebraic actions that characterize Hausdorffness, topological freeness, and minimality of the associated tight groupoid. We parameterize all closed invariant subspaces of the unit space of our groupoid, and characterize topological freeness of the associated reduction groupoids. We prove that our groupoids are purely infinite whenever they are minimal, and in the topologically free case, we prove that our concrete C*-algebra is always a (possibly exotic) groupoid C*-algebra in the sense that it sits between the full and essential C*-algebras of our groupoid. As an application, we obtain structural results for C*-algebras associated with, for instance, shifts over semigroups, actions coming from commutative algebra, and non-commutative rings.
We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch--Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.
In recent joint work of the authors with Laca, we precisely formulated the notion of partition function in the context of C*-dynamical systems. Here, we compute the partition functions of C*-dynamical systems arising from Toeplitz algebras of graphs, and we explicitly recover graph-theoretic information in terms of C*-dynamical invariants. In addition, we compute the type for KMS states on C*-algebras of finite (reducible) graphs and prove that the extremal KMS states at critical inverse temperatures give rise to type III$_\lambda$ factors. Our starting point is an independent result parameterising the partition functions of a certain class of C*-dynamical systems arising from groupoid C*-algebras in terms of $\beta$-summable orbits.
We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray--von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under $\mathbb{R}$-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the number-theoretic background and of the technical results about congruence monoids.
We study semigroup C*-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. Using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup C*-algebras.
We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta\in[1,2]$, there is a unique KMS$_\beta$ state, and we prove that it is a factor state of type III$_1$. There is a phase transition at $\beta=2:$ For each $\beta\in (2,\infty]$, the set of extremal KMS$_\beta$ states decomposes as a disjoint union over a quotient of a ray class group in which the fibers are extremal traces on certain group C*-algebras associated with the ideal classes. Moreover, in most cases, there is a further phase transition at $\beta=\infty$ in the sense that there are ground states that are not KMS$_\infty$ states. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.
Let $K$ be a number field with ring of integers $R$. Given a modulus $\mathfrak{m}$ for $K$ and a group $\Gamma$ of residues modulo $\mathfrak{m}$, we consider the semi-direct product $R\rtimes R_{\mathfrak{m},\Gamma}$ obtained by restricting the multiplicative part of the full $ax+b$-semigroup over $R$ to those algebraic integers whose residue modulo $\mathfrak{m}$ lies in $\Gamma$, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo $\mathfrak{m}$, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full $ax+b$-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of $R\rtimes R_{\mathfrak{m},\Gamma}$ embeds canonically into the left regular C*-algebra of the full $ax+b$-semigroup. Our methods rely heavily on Li's theory of semigroup C*-algebras.
Each multiplicative real-valued homomorphism on a quasi-lattice ordered monoid gives rise to a quasi-periodic dynamics on the associated Toeplitz C*-algebra; here we study the KMS equilibrium states of the resulting C*-dynamical system. We show that, under a nondegeneracy assumption on the homomorphism, there is a critical inverse temperature $\beta_c$ such that at each inverse temperature $\beta \geq \beta_c$ there exists a unique KMS state. Strictly above $\beta_c$, the KMS states are generalised Gibbs states with density operators determined by analytic extension to the upper half-plane of the unitaries implementing the dynamics. These are faithful Type~I states. The critical value $\beta_c$ is the largest real pole of the partition function of the system and is related to the clique polynomial and skew-growth function of the monoid, relative to the degree map given by the logarithm of the multiplicative homomorphism. Motivated by the study of equilibrium states, we give a proof of the inversion formula for the growth series of a quasi-lattice ordered monoid in terms of the clique polynomial as in recent work of Albenque--Nadeau and McMullen for the finitely generated case, and in terms of the skew-growth series as in recent work of Saito. Specifically, we show that $e^{-\beta_c}$ is the smallest pole of the growth series and thus is the smallest positive real root of the clique polynomial. We use this to show that equilibrium states in the subcritical range can only occur at inverse temperatures that correspond to roots of the clique polynomial in the interval $(e^{-\beta_c},1)$, but we are not aware of any examples in which such roots exist.