scholarly journals Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

2011 ◽  
Vol 109 (1) ◽  
pp. 71
Author(s):  
Alexander Pavlov ◽  
Ulrich Pennig ◽  
Thomas Schick
Keyword(s):  
Strict Topology ◽  
Locally Compact ◽  
Compact Spaces ◽  
Equivalent Definition ◽  
C Algebras ◽  
Definition Of ◽  
Locally Compact Spaces

Quasi-multipliers for a Hilbert $C^*$-bimodule $V$ were introduced by L. G. Brown, J. A. Mingo, and N.-T. Shen [3] as a certain subset of the Banach bidual module $V^{**}$. We give another (equivalent) definition of quasi-multipliers for Hilbert $C^*$-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over $C^*$-algebras, provided these $C^*$-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule $l_2(A)$ and for bimodules of sections of Hilbert $C^*$-bimodule bundles over locally compact spaces.

scholarly journals RECOVERING THE BOUNDARY PATH SPACE OF A TOPOLOGICAL GRAPH USING POINTLESS TOPOLOGY

2020 ◽  
pp. 1-17
Author(s):  
GILLES G. DE CASTRO
Keyword(s):  
Path Space ◽  
Topological Graph ◽  
Topological Graphs ◽  
Locally Compact ◽  
C Algebra ◽  
Compact Spaces ◽  
Compact Topology ◽  
Boundary Path ◽  
Definition Of ◽  
Locally Compact Spaces

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.

Strict Topology on Paracompact Locally Compact Spaces

1977 ◽  
Vol 29 (1) ◽  
pp. 216-219 ◽  
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Compact Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Complex Numbers ◽  
Strict Topology ◽  
Locally Compact ◽  
Compact Spaces ◽  
Locally Compact Space ◽  
Locally Compact Spaces ◽  
Locally Convex

In this paper, X denotes a Hausdorff paracompact locally compact space, E a Hausdorff locally convex space over K, the field of real or complex numbers (we call the elements of K scalars), a filtering upwards family of semi-norms on E generating the topology of E, Cb(X) the space of all continuous scalar-valued funcions on X, and Cb(X, E) the space of all continuous, bounded E-valued functions.

On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

2005 ◽  
Vol 57 (6) ◽  
pp. 1121-1138 ◽  
Author(s):  
Michael Barr ◽  
R. Raphael ◽  
R. G. Woods
Keyword(s):  
Locally Compact ◽  
Compact Spaces ◽  
Open Questions ◽  
Tychonoff Spaces ◽  
Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

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