Classical Invariants for Global Actions and Groupoid Atlases

A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and groupoid atlases, such as the strong fundamental groupoid, the weak and strong nerves, classifying spaces and homology groups. We relate all these new invariants to classical constructions in topological spaces, simplicial complexes and simplicial sets. This way we obtain new combinatorial formulations of classical and non classical results in terms of groupoid atlases.      

Representations of orbifold groupoids

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an almost free action of a proper bundle of topological groups.     

RANDOMLY CONNECTED DYNAMICAL SYSTEMS ON BANACH SPACES

We give sufficient conditions for asymptotic stability of Markov operators governing the evolution of measures due to the action of randomly chosen dynamical systems on Banach spaces. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for a semigroup generated by the considered systems.     

Ramsey-Milman phenomenon,Urysohn metric spaces,and extremely amenable groups

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso of isometries of Urysohn’s universal complete separable metric space , equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.     

Partial category actions on sets and topological spaces

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schröder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.     

Tychonoff's theorem for locally compact spaces and an elementary approach to the topology of path spaces

The path spaces of a directed graph play an important role in the study of graph -algebras. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology.

     

Groupoid cohomology and extensions

We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.

     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Transitive Isometry Groups of Aspheric Riemannian Manifolds

We consider the isometry groups of Riemannian solvmanifolds and also study a wider class of homogeneous aspheric Riemannian spaces. We clarify the topological structure of these spaces (Theorem 1). We demonstrate that each Riemannian space with a maximally symmetric metric admits an almost simply transitive action of a Lie group with triangular radical (Theorem 2). We apply this result to studying the isometry groups of solvmanifolds and, in particular, solvable Lie groups with some invariant Riemannian metric.     

Invariant means on double coset spaces

In this paper we study invariant means on and amenability of double coset spaces. We prove the amenability of Gelfand pairs. As an application we prove a stability theorem for a functional equation related to spherical functions.     

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.     

Amenable unitary representations of locally compact groups

Summary A notion of amenability for an arbitrary unitary group representation is introduced. This unifies and generalizes the notions of amenable homogeneous spaces and of inner-amenable groups. Amenable locally compact groups are characterized by the amenability of all their unitary representations. Amenable representations are characterized by several properties which are operator theoretic analogues of properties characterizing amenable groups. We give a generalization to arbitrary representations of Hulanicki-Reiter theorem. This is used in order to describe the amenable representations of the groups with Kazhdan property (T).     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^{?}$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.     

Exponential coordinates and regularity of groupoid heat kernels

We prove that on an asymptotically Euclidean boundary groupoid, the heat kernel of the Laplacian is a smooth groupoid pseudo-differential operator.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

The index of geometric operators on Lie groupoids

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the “topological side” of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.     

Multiple traces boundary integral formulation for Helmholtz transmission problems

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in ??subdomains??) that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise $\widetilde{H}^{-1/2}$ -distributions over each associated interface. Through the use of interior Calderón projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions?? projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.     

Smale空间上的广群C~*-代数的迹

本文证明由拓扑混合的Smale空间上的渐进等价关系定义的广群C*-代数及其相应的Ruelle代数有唯一的迹态;在拓扑可迁的情形下,证明此C*-代数的迹态构成了一个单形,此单形顶点的个数等于“Smale谱分解”中基本空间的个数,单形的重心是该C*-代数的唯一的αa-不变迹态;此回答了I.Putnam的一个猜测.     

The natural rearrangement invariant structure on tensor products

We prove that the only rearrangement invariant (r.i.) spaces for which there exists a crossnorm verifying that the tensor product of these spaces preserves the “natural” r.i. space structure, in the sense that it makes the multiplication operator B a topological isomorphism, are the Lp spaces.     

Q ℒ,q-Scales in a Hyperbolic Setting

Q_p -scales arise in complex analysis as an interpolation scale between BMO, Bloch and Dirichlet spaces. They were generalized to the n-dimensional case by means of the conformal group of the unit ball and a modified fundamental solution of the Laplacian; however, this operator is no longer invariant under the action of group in consideration. In this talk we propose an approach to Q _ℒ,q-scales for homogeneous hyperbolic manifolds using a fundamental solution for the (α -homogeneous) hyperbolic Dirac operator based on a spherical Radon transform. We present also some properties of this scales. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)