Thierry Giordano

University of Ottawa · Department of Mathematics and Statistics

A family of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite–Birkhoff–Taylor methods of order p = 5,6,…,14, that we denote by CPHBTRK4(d,s,p), with nonnegative coefficients are constructed by casting s-stage Runge–Kutta methods of order 4 with Taylor methods of order d. The constructed CPHBTRK4 methods are implem...

The appearance of counterexamples to (EC) in the first half of the 2000s forced a re-evaluation of the classification program Elliott–Toms [14]. The fact that so many natural classes of C*-algebras do satisfy (EC) made it impossible to believe that instances of its confirmation were somehow random. There had to be an underlying principle governing...

This is the first of three sections devoted to examples of group actions.

The absorption theorem is the main result in this chapter and is the key tool we will use to classify up to orbit equivalence minimal AF-equivalence relations and étale equivalence relations associated to minimal actions on the Cantor set of finitely generated abelian groups. This key result shows that a “small” extension of a minimal AF-equivalenc...

In Definition 20.4.33, we introduced the invariant Dm(X,\(\mathcal{R}\)) for an étale equivalence relation \(\mathcal{R}\) on a totally disconnected, compact Hausdorff space X, and showed in Proposition 20.4.40 that it is an orbit equivalence invariant. In this chapter, we will show that this invariant is complete for the class of AF equivalence re...

Complementing the internal finite modelling property of local finiteness is the external property of local embeddability into finite groups, which is the topological (i.e., purely group-theoretic, since our groups are discrete) analogue of soficity. The group G is said to be LEF (locally embeddable into finite groups) if for every finite set F ⊆ G...

These notes are an introduction to group actions on C*-algebras and their crossed products, primarily by discrete groups and with emphasis on situations in which the crossed products are simple and at least close to the class of C*-algebras expected to be classifiable in the sense of the Elliott program. They are aimed at graduate students who have...

The notion of full group was introduced in 1959 by H. Dye in his study of orbit equivalence of measured dynamical systems [19] and [20].

As discussed in Section 14.1, for discrete groups the basic idea of internal measuretheoretic finite approximation is captured by the Følner set characterization of amenability. At the same time we can view Følner sets as furnishing external finite approximations in the following way. Let F be a nonempty finite subset of a discrete group G.

In this part, we give an introduction to large subalgebras of C*-algebras and some applications. Much of the text of this part is taken directly from [215], which is a survey of applications of large subalgebras based on lectures given at the University of Wyoming in the summer of 2015. That survey assumes much more background than these notes (it...

The topological analogue of amenability for discrete groups is local finiteness. In contrast to the setting of C*-algebras, which we will turn to below, for discrete groups the topological notion of perturbation is trivial and thus, unlike the combinatorial measure-theoretic viewpoint, does not give us anything new beyond the merely group-theoretic...

The notion of amenability in its most basic combinatorial sense captures the idea of internal finite approximation from a measure-theoretic perspective. It plays a pivotal role not only in combinatorial and geometric group theory but also in the theory of operator algebras through its various linear manifestations like hyperfiniteness, semidiscrete...

The main goal of this chapter is the presentation of the Bratteli–Vershik model developed by R. Herman, I.F. Putnam and C.F. Skau in their remarkable paper [46].

Our main interest is in structural results for crossed products. We want simplicity, but we really want much more than that. We particularly want theorems showing that certain crossed products are in classes of C*-algebras known to be covered by the Elliott classification program, so that the crossed product can be identified up to isomorphism by c...

In this section, we discuss free and essentially free minimal actions of countable discrete groups on compact metric spaces, with emphasis on minimal homeomorphisms (actions of ℤ). We give two simplicity proofs, using very different methods. One works for free minimal actions, and the method gives further information, as well as some information wh...

In the first part of this chapter we will present Jewett–Krieger type realization results of an ergodic dynamical system by a Cantor minimal system in a prescribed orbit equivalence class. Nic Ormes proved them in his thesis and they are published in [71].

Here, we introduce the basic theory of the Cuntz semigroup. Any proofs not contained in these notes can be found in Ara–Brown–Guido–Lledo–Perera–Toms [1], but no originality is claimed there. Most of the results here are due to Cuntz [9], Kirchberg–Rørdam [17], and Rørdam [26]. We assume that all C*-algebras are separable unless otherwise noted.

The main focus of these notes is the structure of certain kinds of crossed products. The C*-algebra of a group is a special case of a crossed product—it comes from the trivial action of the group on ℂ—but not one of the ones we are mainly concerned with. We devote this section and Section 9.3 to group C*-algebras anyway, in order to provide an intr...

In this section we take some of the salient results of the preceding section and show more or less how they are proved. We shall take for granted, however, Winter’s locally finite nuclear dimension theorem; a full proof is well beyond the scope of these notes.

In measurable dynamics, the study of orbit equivalence, initiated by Dye [19], was developed by Krieger [56], Ornstein–Weiss [73] and Connes–Feldman–Weiss [13] among many others in the amenable case. The strategy of their proofs consisted of showing that any amenable measurable equivalence relation is orbit equivalent to a hyperfinite measurable eq...

What does the Cuntz semigroup really look like? We will see later that the question is out of reach in any reasonable sense without imposing some conditions on your C*-algebra. For this section, we restrict our attention to unital simple separable C*-algebras which have strict comparison of positive elements. In light of Example 2.0.5, we will also...

In the first section of this chapter, we will first recall the definition and the first properties of étale equivalence relations.We restrict our presentation to the notions we will need in the next chapters; for more details see, for example, [74, 78, 80]. In the second section, we recall the definitions of isomorphism and orbit equivalence of éta...

To any continuous eigenvalue of a Cantor minimal system $(X,\,T)$, we associate an element of the dimension group $K^0(X,\,T)$ associated to $(X,\,T)$. We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous...

In 1976, D. Voiculescu proved that every separable unital sub-C*-algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see reference), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a $\sigma$-unital C*-algebra. In this note, we answer Pedersen's question for a stab...

We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general l...

The contractivity-preserving 2- and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2- and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17--39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contra...

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal $\Z^{d}$-odom...

A residual error estimator is proposed for the energy norm of the error for a scalar reaction-diffusion problem and for the monodomain model used in cardiac electrophysiology. The problem is discretized using $P_1$ finite elements in space, and the backward difference formula of second order (BDF2) in time. The estimator for space makes use of anis...

In this paper we show that the natural action of the symmetric group acting on the product space {0, 1 }^N endowed with a symmetric measure is approximately transitive. We also extend the result to a larger class of probability measures.

New optimal, contractivity-preserving (CP), \(d\)-derivative, 2- and 3-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by \(HBO(d,k,p)\), \(k=2,3\), with nonnegative coefficients are constructed for solving nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bounds \(p_u\) of order \(p...

Let $U$ and $V$ be open subsets of the Cantor set with finite disjoint complements, and let $h:U\to V$ be a homeomorphism with dense orbits. Building from the ideas of Herman, Putnam, and Skau, we show that the partial action induced by $h$ can be realized as the Vershik map on a Bratteli diagram, and that any two such diagrams are equivalent.

An element based adaptation method is developed for an anisotropic a posteriori error estimator. The adaptation does not make use of a metric, but instead equidistributes the error over elements using local mesh modifications. Numerical results are reported, comparing with three popular anisotropic adaptation methods currently in use. It was found...

Variable-step (VS) \(4\)-stage \(k\)-step Hermite--Birkhoff (HB) methods of order \(p=(k+1)\), denoted by HB\((p)\), are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a two-step diagonally implicit \(4\)-stage Runge--Kutta method of order 3 (TSDIRK3) for solving stiff ordinary differential equations. The main reas...

New optimal, contractivity-preserving (CP), explicit, d-derivative, k-step Hermite–Obrechkoff series methods of order p up to \(p=20\), denoted by CP HO(d, k, p), with nonnegative coefficients are constructed. These methods are used to solve nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bound \(p_u\) of order...

For any adic transformation $T$ defined on the path space $X$ of an ordered Bratteli diagram, endowed with a Markov measure $\mu$, we construct an explicit dimension space (which corresponds to a matrix values random walk on $\mathbb{Z}$) whose Poisson boundary can be identified as a $\mathbb{Z}$-space with the dynamical system $(X,\mu,T)$. We give...

In this paper we analyze approximately finite dimensional von Neumann algebras obtained as weak closure of fixed point algebras under xerox type action of compact groups on UHF algebras. We reduce our analysis to the case of diagonal xerox type actions. The von Neumann algebras that we study are identified with von Neumann algebras associated to ce...

Variable-step (VS) $$3$$3-stage Hermite–Birkhoff (HB) methods HB$$(p=k+1)$$(p=k+1) of order $$p=8$$p=8 and 9 are constructed as a combination of linear $$k$$k-step methods of order $$(p-2)$$(p-2) and a diagonally implicit one-step $$3$$3-stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor...

Let (A,G,\alpha) be a partial dynamical system. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A xr G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition---automatic when A is abelian---we show that A xr G is purely i...

In 1955 Dye proved that two von Neumann factors not of type I_2n are isomorphic (via a linear or a conjugate linear *-isomorphism) if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for C*-algebras. We show that the topological general linear group is a classifying invariant for simple, unital AH-algebras...

In this note, we derive some consequences of the von Neumann algebra uniqueness theorems developed in a previous paper (see arXiv:1207.6741v1). In particular, 1) we solvein a paper of Futamura, Kataoka, and Kishimoto, by proving that if A is a separable simple nuclear C*-algebra and for \pi_1 and \pi_2 are type III representations of A on a separab...

The main result of this paper is a characterization of properly infinite injective von Neumann algebras and of nuclear C*-algebras by using a uniqueness theorem, based on generalizations of Voiculescu's famous Weyl-von Neumann theorem.

Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. We show that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their K0K0-ordered groups are isomorphic. Also, using Gong and Dadarlatʼs classification theore...

Optimal, 7-stage, explicit, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) methods of orders 4 to 8 with nonnegative coefficients are constructed by combining linear k-step methods with a 7-stage Runge–Kutta (RK) method of order 4. Compared to Huang’s hybrid methods of the same order, the new methods generally have larger effective SSP coe...

We describe the enveloping C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of Z on the Cantor set, the enveloping C*-algebra is an AF-alg...

Strong-stability-preserving (SSP) time-discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws. A collection of SSP explicit 3-stage Hermite–Birkhoff methods of orders 3 to 7 with nonnegative coefficients are constructed as k-step analogues of third-order R...

We consider a minimal, free action, ϕ, of the group Z d on the Cantor set X, for d ≥ 1. We introduce the notion of small positive co-cycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani-Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the a...

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and ℤ d -actions.

We describe the envelope C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of Z on the Cantor set, the envelope C*-algebra is an AF-algebra...

We reformulate matrix-valued random walks and their associated group actions in terms of dimension groups, suitably modified to deal with measure-theoretic classification. This leads naturally to a notion of rank denoted AT(n), for integers n (approximately transitive, that is, AT, actions constitute the rank one situation). This yields wide classe...

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case--when...

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and Z^d-actions.

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the produ...

We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper....

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the unifo...

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups t...

We answer a question of R. Grigorchuk by showing the following characterization: for a countable group Γ to be amenable, it is necessary and sufficient that any continuous action of Γ on the Cantor set has an invariant probability measure. The proof uses an easy variation of a classical result of Alexandroff and Urysohn: any metrisable Γ-space is a...

We answer a question of R. Grigorchuk by showing the following characterization: for a countable group Γ to be amenable, it is necessary and sufficient that any continuous action of Γ on the Cantor set has an invariant probability measure. The proof uses an easy variation of a classical result of Alexandroff and Urysohn: any metrisable Γ-space is a...

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebra which is isomorphic to a unital subalgebra of the centre of M(A), the multiplier algebra of A. Letting Ω = Ĉ, so that we may write C = C(Ω), we call A a C(Ω)-algebra (following Blanchard [7]). Suppose that B is another C(Ω)-algebra, then we form A⊗C B, the algebraic tensor produ...

The equivalence between different characterizations of amenable actions of a locally compact group is proved. In particular, this answers a question raised by R. J. Zimmer in 1977.

A structure theorem is established for amenable actions of a countable discrete group.

A real AF C *-algebra is the norm closure of a direct limit of finite dimensional real C *-algebras (with real *-algebra maps). When we use the unadorned “AF C *-algebra”, we mean the usual complex version. Let R be a simple AF C *-algebra such that K 0 ( R ) is free of rank 2 or 3. The problem is to find (up to Morita equivalence) all real AF C *-...

It is proved, using Krieger's theorem, that ITPFI's of bounded type are ITPFI2. This answers a question asked by E. J. Woods.

A factor M, isomorphic to its tensor square, whose Sakai flip σϵ Aut(M ⊗ M) is approximately inner, has a flow of weights with pure point spectrum.

Let B be a σ-finite von Neumann factor of type II 1 or III and let σ be an involutory *-antiautomorphism of B . We consider U ( B ) the unitary group of B and its subgroup G = { g ∈U ( B ) | σ( g ) = g *}, which are unitary classical groups. In this paper, we prove that G has a unique non trivial normal subgroup, which is its centre { ±1 }.

This paper considers minimal, free actions of the group ℤ 2 on a compact, totally disconnected space having no isolated points. Under a hypothesis involving the existence of sufficiently many ‘small, positive’ cocycles, a procedure is given for finding a nested sequence of compact, open sub-equivalence relations of the orbit relation. It is shown t...

New optimal strong-stability-preserving Hermite– Birkhoff (SSP HB) methods, HB(k, s, p), of order p = 4, 5, . . . , 12, are constructed by combining k-step methods of order p = 1, 2, ...,9 and s-stage explicit Runge–Kutta (RK) methods of order 4, where s = 4, 5, . . . , 10. These methods are well suited for solving discretized hyperbolic PDEs by th...

The project is an ongoing one; the first paper appeared in 1995. The program is to classify, up to orbit equivalence, group actions and, more generally etale equivalence relations on Cantor speaces. During the RIT period at BIRS, we worked on the case of minimal, free Z 2 actions on Cantor sets. The following set of notes was written by Ian Putnam...