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Antonios Manoussos - One of the best experts on this subject based on the ideXlab platform.
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the group of isometries of a locally Compact Metric Space with one end
Topology and its Applications, 2010Co-Authors: Antonios ManoussosAbstract:In this note we study the dynamics of the natural evaluation action of the group of isometries G of a locally Compact Metric Space (X, d) with one end. Using the notion of pseudo-components introduced by S. Gao and A.S. Kechris we show that X has only finitely many pseudo-components exactly one of which is not Compact and G acts properly on this pseudo-component. The complement of the non-Compact component is a Compact subset of X and G may fail to act properly on it. (C) 2010 Elsevier B.V. All rights reserved.
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The group of isometries of a locally Compact Metric Space with one end
arXiv: General Topology, 2009Co-Authors: Antonios ManoussosAbstract:In this note we study the dynamics of the natural evaluation action of the group of isometries $G$ of a locally Compact Metric Space $(X,d)$ with one end. Using the notion of pseudo-components introduced by S. Gao and A. S. Kechris we show that $X$ has only finitely many pseudo-components of which exactly one is not Compact and $G$ acts properly on. The complement of the non-Compact component is a Compact subset of $X$ and $G$ may fail to act properly on it.
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On the action of the group of isometries on a locally Compact Metric Space
arXiv: General Topology, 2008Co-Authors: Antonios ManoussosAbstract:In this short note we give an answer to the following question. Let X be a locally Compact Metric Space with group of isometries G. Let {gi} be a net in G for which gix converges to y, for some x;y ∈ X. What can we say about the convergence of {gi}? We show that there exist a subnet {gj } of {gi} and an isometry f : Cx → X such that gj converges to f pointwise on Cx and f(Cx) = Cy, where Cx and Cy denote the pseudo-components of x and y respectively. Applying this we give short proofs of the van Dantzig{van der Waerden theorem (1928) and Gao{Kechris theorem (2003). 1. The main result and some applications A few words about the notation we shall be using. In what follows, X will denote a locally Compact Metric Space with group of isometries G. If we endow G with the topology of pointwise convergence then G is a topological group (2, Ch. X, §3.5 Corollary). On G there is also the topology of uniform convergence on Compact subsets which is the same as the Compact-open topology. In the case of a group of isometries these topologies coincide with the topology of pointwise convergence, and the natural action of G on X with (g;x) 7→ g(x), g ∈ G, x ∈ X, is continuous (2, Ch. X, §2.4 Theorem 1 and §3.4 Corollary 1). For F ⊂ G, let K(F ) := {x ∈ X | the set Fx has Compact closure in X}. The sets K(F ) are clopen (6, Lemma 3.1).
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On the group of isometries on a locally Compact Metric Space.
Journal of Lie Theory, 2003Co-Authors: Antonios Manoussos, Polychronis StrantzalosAbstract:In the present paper we study conditions under which the group of isometries on a locally Compact Metric Space is locally Compact, or acts properly.
Frederic Latremoliere - One of the best experts on this subject based on the ideXlab platform.
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Quantum locally Compact Metric Spaces
Journal of Functional Analysis, 2012Co-Authors: Frederic LatremoliereAbstract:Abstract We introduce the notion of a quantum locally Compact Metric Space, which is the noncommutative analogue of a locally Compact Metric Space, and generalize to the non-unital setting the notion of quantum Metric Spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, Compact quantum Metric Spaces and locally Compact Metric Spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum Metric Space in the non-unital setup and offers important insights into noncommutative geometry for non-Compact quantum Spaces.
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Quantum Locally Compact Metric Spaces
arXiv: Operator Algebras, 2012Co-Authors: Frederic LatremoliereAbstract:We introduce the notion of a quantum locally Compact Metric Space, which is the noncommutative analogue of a locally Compact Metric Space, and generalize to the nonunital setting the notion of quantum Metric Spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, as well as Compact quantum Metric Spaces and locally Compact Metric Spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum Metric Space in the nonunital setup and offers important insights into noncommutative geometry for non Compact quantum Spaces.
P Veeramani - One of the best experts on this subject based on the ideXlab platform.
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on the equivalence of the mizoguchi takahashi fixed point theorem to nadler s theorem
Applied Mathematics Letters, 2009Co-Authors: Anthony A Eldred, J Anuradha, P VeeramaniAbstract:Abstract In this work, we prove that on a Metrically convex complete Metric Space, the Mizoguchi–Takahashi theorem is equivalent to Nadler’s theorem. Also, we obtain its equivalence on a Compact Metric Space.
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On the equivalence of the Mizoguchi–Takahashi fixed point theorem to Nadler’s theorem
Applied Mathematics Letters, 2009Co-Authors: Anthony A Eldred, J Anuradha, P VeeramaniAbstract:Abstract In this work, we prove that on a Metrically convex complete Metric Space, the Mizoguchi–Takahashi theorem is equivalent to Nadler’s theorem. Also, we obtain its equivalence on a Compact Metric Space.
Debashish Goswami - One of the best experts on this subject based on the ideXlab platform.
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Existence and examples of quantum isometry groups for a class of Compact Metric Spaces
Advances in Mathematics, 2015Co-Authors: Debashish GoswamiAbstract:We formulate a definition of isoMetric action of a Compact quantum group (CQG) on a Compact Metric Space, generalizing Banica's definition for finite Metric Spaces. For Metric Spaces (X,d) which can be isoMetrically embedded in some Euclidean Space, we prove the existence of a universal object in the category of the Compact quantum groups acting isoMetrically on (X,d). In fact, our existence theorem applies to a larger class, namely for any Compact Metric Space (X,d) which admits a one-to-one continuous map f:X→Rn for some n such that d0(f(x),f(y))=ϕ(d(x,y)) (where d0 is the Euclidean Metric) for some homeomorphism ϕ of R+. As concrete examples, we obtain Wang's quantum permutation group Sn+ and also the free wreath product of Z2 by Sn+ as the quantum isometry groups for certain Compact connected Metric Spaces constructed by taking topological joins of intervals in [13].
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Existence and examples of quantum isometry group for a class of Compact Metric Spaces
arXiv: Operator Algebras, 2012Co-Authors: Debashish GoswamiAbstract:We formulate a definition of isoMetric action of a Compact quantum group (CQG) on a Compact Metric Space, generalizing Banica's definition for finite Metric Spaces. For Metric Spaces $(X,d)$ which can be isoMetrically embedded in some Euclidean Space, we prove the existence of a universal object in the category of the Compact quantum groups acting isoMetrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any Compact Metric Space $(X,d)$ which admits a one-to-one continuous map $f : X \raro \IR^n$ for some $n$ such that $d_0(f(x),f(y))=\phi(d(x,y))$ (where $d_0$ is the Euclidean Metric) for some homeomorphism $\phi$ of $\IR^+$. As concrete examples, we obtain Wang's quantum permutation group $\cls_n^+$ and also the free wreath product of $\IZ_2$ by $\cls_n^+$ as the quantum isometry groups for certain Compact connected Metric Spaces constructed by taking topological joins of intervals in \cite{huang1}.
Tatsuji Kawai - One of the best experts on this subject based on the ideXlab platform.
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Point-free characterisation of Bishop Compact Metric Spaces
Journal of Logic and Analysis, 2017Co-Authors: Tatsuji KawaiAbstract:We give a point-free characterisation of Bishop Compact Metric Spaces in terms of formal topology. We show that the notion of overt Compact enumerably completely regular formal topology is a point-free counterpart of that of Bishop Compact Metric Space. Specifically, a formal topology is isomorphic to an overt Compact enumerably completely regular formal topology if and only if it is isomorphic to the image of a Compact Metric Space under the localic completion of Metric Spaces into formal topologies. The result is obtained in Bishop constructive mathematics with the axiom of Dependent Choice.
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A point-free characterisation of Bishop locally Compact Metric Spaces
Journal of Logic and Analysis, 2017Co-Authors: Tatsuji KawaiAbstract:We give a characterisation of Bishop locally Compact Metric Spaces in terms of formal topology. We identify inhabited enumerably locally Compact regular formal topologies as point-free counterpart of Bishop locally Compact Metric Spaces. Specifically, we show that a formal topology is isomorphic to an inhabited enumerably locally Compact regular formal topology if and only if it is isomorphic to the localic completion of a Bishop locally Compact Metric Space. The result is obtained in Bishop constructive mathematics with the Dependent Choice.