松井 宏樹

<p lang="fr">&lt;p style='text-indent:20px;'&gt;When a pair of étale groupoids &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on totally disconnected spaces are related in some way, we discuss the difference of their homology groups. More specifically, we treat two basic situations. In the subgroupoid situation, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is assumed to be an open regular subgroupoid of &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In the factor groupoid situation, we assume that &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a quotient of &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \mathcal{G} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the factor map &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \mathcal{G}\to\mathcal{G}' $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is proper and regular. For each, we show that there exists a long exact sequence of homology groups. We present examples which arise from SFT groupoids and hyperplane groupoids.&lt;/p&gt;</p>

<title>Abstract</title> Toward the complete classification of poly-${\mathbb{Z } }$ group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly-${\mathbb{Z } }$ groups of Hirsch length not greater than three on unital Kirchberg algebras up to $KK$-trivial cocycle conjugacy.

In this paper, we will study representations of the continuous full group Gamma(A) of a one-sided topological Markov shift (X-A, sigma(A)) for an irreducible matrix A with entries in {0, 1} as a generalization of Higman-Thompson groups V-N, 1 &lt; N is an element of N. We will show that the group Gamma(A) can be represented as a group Gamma(tab)(A) A of matrices, called A-adic tables, with entries in admissible words of the shift space X-A, and a group Gamma(PL)(A) A of right continuous piecewise linear functions, called A-adic PL functions, on [0, 1] with finite singularities.

Two conjectures about homology groups, K-groups and topological full groups of minimal Bale groupoids on Cantor sets are formulated. We verify these conjectures for many examples of etale groupoids including products of etale groupoids arising from one-sided shifts of finite type. Furthermore, we completely determine when these product groupoids are mutually isomorphic. Also, the abelianization of their topological full groups is computed. They are viewed as generalisations of the higher dimensional Thompson groups. (C) 2016 Elsevier Inc. All rights reserved.

For continuously orbit equivalent one-sided topological Markov shifts. (X-A, sigma(A)) and. (X-B, sigma(B)), their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups ((H) over bar (A), (H) over bar (A)(+)) and ((H) over bar (B), (H) over bar (B)(+)) We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

This is a survey of the recent development of the study of topological full groups of étale groupoids on the Cantor set. Étale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations.Minimal ℤ-actions, minimal ℤN-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so on. Homology groups of étale groupoids, groupoid C∗-algebras and their K-groups are also investigated.

We explore the topological full group [G] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [G] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D([G]) is shown to be simple. The etale groupoid G arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, [G] is thought of as a generalization of the Higman-Thompson group. We prove that [G] is of type F-infinity, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of [G] is calculated and described in terms of the homology groups of G.

We study universal properties of locally compact G-spaces for countable infinite groups G. In particular we consider open invariant subsets of the G-space PG, and their minimal closed invariant subspaces. These are locally compact free G-spaces, and the latter are also minimal. We examine the properties of these G-spaces with emphasis on their universal properties. As an example of our results, we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non-compact Cantor set. (C) 2015 Elsevier Inc. All rights reserved.

We investigate outer symmetries on unital Kirchberg algebras with respect to the Rokhlin property and finite Rokhlin dimension. In stark contrast to the restrictiveness of the Rokhlin property, every such action has Rokhlin dimension at most one. A consequence of these observations is a relationship between the nuclear dimension of an O-infinity-absorbing C*-algebra and its O-2-stabilization. We also give a more direct and alternative approach to this result. Several applications of this relationship are discussed to cover a fairly large class of O-infinity-absorbing C*-algebras that turn out to have finite nuclear dimension.

We consider a crossed product of a unital simple separable nuclear stably finite Z-stable C*-algebra A by a strongly outer cocycle action of a discrete countable amenable group P. Under the assumption that A has finitely many extremal tracial states and Gamma is elementary amenable, we show that the twisted crossed product C*-algebra is Z-stable. As an application, we also prove that all strongly outer cocycle actions of the Klein bottle group on Z are cocycle conjugate to each other. This is the first classification result for actions of non-abelian infinite groups on stably finite C*-algebras.

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that A is nuclear, quasidiagonal, and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers-Sakai conjecture.

We introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman-Thompson groups by using the topological full groups of the groupoids defined by beta-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the beta-expansion of 1 is finite or ultimately periodic. We also classify them by a number-theoretical property of beta.

Let A, B be square irreducible matrices with entries in {0, 1}. We will show that if the one-sided topological Markov shifts (XA, σA) and (XB, σB) are continuously orbit equivalent, then the two-sided topological Markov shifts (X¯A, σA) and (X¯B, σB) are flow equivalent, and hence det(id-A) = det(id-B ). As a result, the one-sided topological Markov shifts (XA, σA) and (XB, σB) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras OA and OB are isomorphic and det(id-A) = det(id-B).

We prove that commutator subgroups of topological full groups arising from minimal subshifts have exponential growth. We also prove that the measurable full group associated to the countable, measure-preserving, ergodic and hyperfinite equivalence relation is topologically generated by two elements.

For any unital separable simple infinite-dimensional nuclear C (au)-algebra with finitely many extremal traces, we prove that -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear C (au)-algebra with tracial rank zero is approximately divisible, and hence is -absorbing.

We consider a certain class of unital simple stably finite C*-algebras which absorb the Jiang-Su algebra tensorially. Under a mild assumption, we show that the crossed product of a C*-algebra in this class by a strongly outer action of or a finite group is -stable. As an application, we also prove that all strongly outer actions of on are mutually cocycle conjugate.

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H-0 if and only if there exists an element in the topological full group that maps one to the other. It is also shown that a natural homomorphism, called the index map, from the topological full group to H-1 is surjective and any element of the kernel can be written as a product of four elements of finite order. In particular, the index map induces a homomorphism from H-1 to K-1 of the groupoid C*-algebra. Explicit computations of homology groups of AF groupoids and etale groupoids arising from subshifts of finite type are also given.

We prove that all strongly outer Z(N)-actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable Z(N)-actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.

We classify unital monomorphisms into certain simple Z-stable C*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C*-algebra, or any unital simple separable nuclear Z-stable C*-algebra satisfying the UCT such that C circle times B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C*-algebra such that A circle times B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C*-algebra whose projections separate traces and whose extremal traces are finitely many. (C) 2010 Elsevier Inc. All rights reserved.

We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z(2)-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.

We classify a large class of Z(2)-actions on the Kirchberg algebras employing the Kasparov group KK1 as the space of classification invariants. (C) 2009 Elsevier Inc. All rights reserved.

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 a"currency sign (d) -actions.

We prove that a 'small' extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a 'small' extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2]. It is needed for the study of orbit equivalence of minimal Z(d)-systems for d &gt; 2 [GMPS3], in a similar way as the result in [GMPS2] was needed (and sufficient) for the study of minimal Z(2)-systems [GMPS1].

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 Y is a finite set-this result is highly non-trivial. The result itself-called the absorption theorem-is a powerful and crucial tool for the study of the orbit structure of minimal Z(n)-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys. 24 (2004), 441-475]. However, we shall need a few key results from the above paper in order to prove the absorption theorem.

We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z(2) on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z(2) on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy. (c) 2008 Elsevier Inc. All rights reserved.

We will show that any two outer actions of Z(N) on O(2) are cocycle conjugate. (C) 2007 Elsevier Inc. All rights reserved.

We say that a C*-algebra A is approximately square root closed if any normal element in A can be approximated by a square of a normal element in A. We study when A is approximately square root closed, and have an affirmative answer for AI-algebras, Goodearl-type algebras over the torus, purely infinite simple unital C*-algebras, etc.

Let G be a finite abelian group. We will consider a skew product extension of a product of two Cantor minimal Z-systems associated with a G-valued cocycle. When G is non-cyclic and the cocycle is non-degenerate, it will be shown that the skew product system has torsion in its coinvariants.

We will show that any extension of a product of two Cantor minimal Z-systems is affable in the sense of Giordano, Putnam and Skau.

We will show that an equivalence relation on a Cantor set arising from a two-dimensional substitution tiling by polygons is affable in the sense of Giordano, Putnam and Skau.

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.

Let X be the Cantor set and phi be a minimal homeomorphism on X x T. We show that the crossed product C*-algebra C* (X x T, phi) is a simple AT-algebra provided that the associated cocycle takes its values in rotations on T. Given two minimal systems (X x T, phi) and (Y x T, psi) such that phi and psi arise from cocycles with values in isometric homeomorphisms on T, we show that two systems are approximately K-conjugate when they have the same K-theoretical information.

Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product C*-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via K-theory and C*-algebras. For example, it is shown that two Cantor minimal systems are approximately tau-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately K-conjugate if and only if the corresponding crossed product C*-algebras have the same scaled ordered K-theory. Consequently, two Cantor minimal systems are approximately K-conjugate if and only if the associated transformation C*-algebras are isomorphic. Incidentally, this approximate K-conjugacy coincides with Giordano, Putnam and Skau's strong orbit equivalence for the Cantor minimal systems.

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if the set of invariant measures of the system come from the associated Cantor minimal system. In the case that cocycles take values in the rotation group, it is also shown that this condition implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately K-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps C( X x T) onto C( X x T).

H. Lin and the author introduced the notion of approximate conjugacy of dynamical systems. In this paper, we will discuss the relationship between approximate conjugacy and full groups of Cantor minimal systems. An analogue of Glasner-Weiss's theorem will be shown. Approximate conjugacy of dynamical systems on the product of the Cantor set and the circle will also be studied. © 2005 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

We show that there exists a locally compact Cantor minimal system whose topological spectrum has a given Hausdorff dimension.

We will study the number of discontinuities of the orbit cocycles associated with orbit equivalence between Cantor minimal systems.

Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.

We will show that the crossed products of unital simple real rank zero AT algebras by the integers are AF embeddable. This is a generalization of Brown's AF embedding theorem. As an application, we will prove the AF embeddability of crossed product algebras arising from certain minimal dynamical systems induced by two commuting homeomorphisms. (C) 2002 Elsevier Science (USA).

When we have two extensions of a Cantor minimal system which are both one-to-one on at least one orbit, we can construct new Cantor minimal systems called topological joinings. We compute the dimension group of the joining in a special case. As an application, we show that a non-invertible endomorphism can induce the identity map on the dimension group of a Cantor minimal system.

We compute the dimension group of the skew product extension of a Cantor minimal system associated with a finite group valued cocycle. Using it, we study finite subgroups in the commutant group of a Cantor minimal system and prove that a finite subgroup of the kernel of the mod map must be cyclic. Moreover, we give a certain obstruction for finite subgroups of commutant groups to have nonzero intersection to the kernel of mod maps. We also give a necessary and sufficient condition for dimension groups so that the kernel of the mod map can include a finite order element.

Giordano, Putnam and Skau showed that the transformation group C*-algebra arising from a Cantor minimal system is an AT-algebra, and classified it by its K-theory. For approximately inner automorphisms that preserve C(X), we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result.