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 introduce positive correspondences as right C*-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the C*-correspondence (Enchilada) category as a "retract". Kasparov's KSGNS construction provides a semi-functor from this semi-category onto the C*-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz-Pimsner algebras, and we describe classes of examples arising from projections on C*-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index. We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences.
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 considered by Kakariadis, Katsoulis, Laca, and X. Li, and provides an elementary proof of a prominent result of Sehnem. As an application, we resolve a question of Li by identifying the C*-envelope of the operator algebra arising from a groupoid-embeddable category.
We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of C*-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular C*-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant *-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for C*-correspondences.
We determine the primitive ideal space and hence the ideal lattice of a large class of separable groupoid C*-algebras that includes all 2-graph C*-algebras. A key ingredient is the notion of harmonious families of bisections in etale groupoids associated to finite families of commuting local homeomorphisms. Our results unify and recover all known results on ideal structure for crossed products of commutative C*-algebras by free abelian groups, for graph C*-algebras, and for Katsura's topological graph C*-algebras.
This paper surveys the recent advances in the interactions between symbolic dynamics and C*-algebras. We explain how conjugacies and orbit equivalences of both two-sided (invertible) and one-sided (noninvertible) symbolic systems may be encoded into C*-algebras, and how the dynamical systems can be recovered from structure-preserving *-isomorphisms of C*-algebras. We have included many illustrative examples as well as open problems.
This note extends and strengthens a theorem of Bates that says that row-finite graphs that are strong shift equivalent have Morita equivalent graph C*-algebras. This allows us to ask whether our stronger notion of Morita equivalence does in fact characterise strong shift equivalence. We believe this will be relevant for future research on infinite graphs and their C*-algebras. We also study insplits and outsplits as particular examples of strong shift equivalences and show that the induced Morita equivalences respect a whole family of weighted gauge actions. We then ask whether strong shift equivalence is generated by (generalised) insplits and outsplits.
*Pete Gautam was an undergraduate student during the time of the research.
We prove a sandwiching lemma for inner-exact locally compact Hausdorff étale groupoids. Our lemma says that every ideal of the reduced C*-algebra of such a groupoid is sandwiched between the ideals associated to two uniquely defined open invariant subsets of the unit space. We obtain a bijection between ideals of the reduced C*-algebra, and triples consisting of two nested open invariant sets and an ideal in the C*-algebra of the subquotient they determine that has trivial intersection with the diagonal subalgebra and full support. We then introduce a generalisation to groupoids of Ara and Lolk's relative strong topological freeness condition for partial actions, and prove that the reduced C*-algebras of inner-exact locally compact Hausdorff étale groupoids satisfying this condition admit an obstruction ideal in Ara and Lolk's sense.
This paper investigates the structure of C*-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The C*-algebras are *-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension of the (groupoid of the) local homeomorphism to be one, and by a result of Guentner, Willett, and Yu, it follows that the nuclear dimension of the C*-algebras is one.
We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterise topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and \(C^*\)-algebras. This significantly generalises recent work of Matsumoto and of the second- and third-named authors.
We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.
To every one-sided shift space \(\mathsf{X}\) we associate a cover \(\tilde{\mathsf{X}}\), a groupoid \(\mathcal{G}_\mathsf{X}\) and a \(C^*\)-algebra \(\mathcal{O}_\mathsf{X}\). We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between \(\mathsf{X}\) and \(\mathsf{Y}\) in terms of isomorphism of \(\mathcal{G}_\mathsf{X}\) and \(\mathcal{G}_\mathsf{Y}\), and diagonal-preserving \(^*\)-isomorphism of \(\mathcal{O}_\mathsf{X}\) and \(\mathcal{O}_\mathsf{Y}\). We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces \(\Lambda_\mathsf{X}\) and \(\Lambda_\mathsf{Y}\) in terms of isomorphism of the stabilized groupoids \(\mathcal{G}_\mathsf{X} \times \mathcal{R}\) and \(\mathcal{G}_\mathsf{Y} \times \mathcal{R}\), and diagonal-preserving \(^*\)-isomorphism of the stabilized \(C^*\)-algebras \(\mathcal{O}_\mathsf{X} \otimes \mathbb{K}\) and \(\mathcal{O}_\mathsf{Y} \otimes \mathbb{K}\). Our strategy is to lift relations on the shift spaces to similar relations on the covers.
Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of \(\mathsf{X}\) from the pair \( (\mathcal{O}_\mathsf{X}, C(\mathsf{X}))\), and the flow equivalence class of \(\Lambda_\mathsf{X}\) from the pair \((\mathcal{O}_\mathsf{X} \otimes \mathbb{K}, C(\mathsf{X}) \otimes c_0)\). In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.
Given an ample groupoid \(G\) with compact unit space, we study the canonical representation of the topological full group \([[G]]\) in the full groupoid \(C^*\)-algebra \(C^*(G)\). In particular, we show that the image of this representation generates \(C^*(G)\) if and only if \(C^*(G)\) admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for \(C^*\)-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of nonamenable and countable groups on the Cantor set.
A one-sided shift of finite type \((\mathsf{X}_A, \sigma_A)\) determines on the one hand a Cuntz–Krieger algebra \(\mathcal{O}_A\) with a distinguished abelian subalgebra \(\mathcal{D}_A\) and a certain completely positive map \(\tau_A\) on \(\mathcal{O}_A\). On the other hand, \((\mathsf{X}_A, \sigma_A)\) determines a groupoid \(\mathcal{G}_A\) together with a certain homomorphism \(\varepsilon_A\) on \(\mathcal{G}_A\). We show that each of these two sets of data completely characterizes the one-sided conjugacy class of \(\mathsf{X}_A\). This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.
The Glasgow Late August Symbolic dynamics, Groups, and Operators Workshop was a workshop held at the University of Glasgow in August 2022 organised by Kevin Aguyar Brix, Chris Bruce, Se-Jin (Sam) Kim, Xin Li, Alistair Miller, and Owen Tanner. The meeting aimed to include especially young mathematicians into the fascinating interactions between group theory, dynamical systems, and operator algebras. The workshop spanned a week (Monday to Friday) and Thursday afternoon had a session dedicated to the discussion and introduction of open problems. This document serves as a rough summary of that discussion.