INVARIANT MEANS AND ACTIONS OF SEMITOPOLOGICAL SEMIGROUPS ON COMPLETELY REGULAR SPACES AND APPLICATIONS

2020 ◽  
pp. 1-12
Author(s):  
KHADIME SALAME
Keyword(s):  
Fixed Point ◽  
Topological Spaces ◽  
Compact Groups ◽  
Locally Convex Spaces ◽  
Locally Compact ◽  
Completely Regular ◽  
Continuous Mappings ◽  
Fixed Point Properties ◽  
Invariant Means ◽  
Locally Convex

In this paper, we extend the study of fixed point properties of semitopological semigroups of continuous mappings in locally convex spaces to the setting of completely regular topological spaces. As applications, we establish a general fixed point theorem, a convergence theorem and an application to amenable locally compact groups.

Invariant means and fixed point properties on completely regular spaces

1977 ◽  
Vol 16 (2) ◽  
pp. 203-212
Author(s):  
Marvin W. Grossman
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Topological Semigroup ◽  
Related Result ◽  
Continuous Functions ◽  
Point Property ◽  
Completely Regular ◽  
Fixed Point Properties ◽  
Bounded Functions ◽  
Invariant Means

Two theorems are presented which characterize the existence of multiplicative left invariant means on a given algebra of unbounded continuous functions on a topological semigroup S in terms of certain common fixed point properties of actions of S on completely regular spaces. Also a lattice formulation of a related result of Theodore Mitchell for the case of bounded functions is shown to be equivalent to a certain common fixed point property on Bauer simplexes.

scholarly journals Fixed points of upper semicontinuous mappings in locally G-convex spaces

1998 ◽  
Vol 58 (3) ◽  
pp. 469-478 ◽  
Author(s):  
George Xian-Zhi Yuan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Topological Spaces ◽  
Locally Convex Spaces ◽  
Upper Semicontinuous ◽  
Special Cases ◽  
Locally Convex ◽  
Convex Spaces

In this paper a new fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values is established in the setting of an abstract convex structure – called a locally G-convex space, which generalises usual convexity such as locally convex H-spaces, locally convex spaces (locally H-convex spaces), hyperconvex metric spaces and locally convex topological spaces. Our fixed point theorem includes corresponding Fan-Glicksberg type fixed point theorems in locally convex H-spaces, locally convex spaces, hyperconvex metric space and locally convex spaces in the existing literature as special cases.

scholarly journals Fixed point theorems for a sum of two mappings in locally convex spaces

1994 ◽  
Vol 17 (4) ◽  
pp. 681-686 ◽  
Author(s):  
P. Vijayaraju
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Locally Convex Spaces ◽  
Continuous Mappings ◽  
Asymptotically Nonexpansive ◽  
Locally Convex ◽  
Convex Spaces

Cain and Nashed generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings as well as the class of contraction mappings. In this paper, we prove by using the same method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in locally convex spaces. These results extend a result of Cain and Nashed.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

scholarly journals On some optimization problems in not necessarily locally convex space

2008 ◽  
Vol 18 (2) ◽  
pp. 167-172
Author(s):  
Ljiljana Gajic
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Theorem ◽  
Convex Space ◽  
Optimization Problems ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Equilibrium Existence ◽  
Cooperative Equilibrium ◽  
Locally Convex

In this note, by using O. Hadzic's generalization of a fixed point theorem of Himmelberg, we prove a non - cooperative equilibrium existence theorem in non - compact settings and a generalization of an existence theorem for non - compact infinite optimization problems, all in not necessarily locally convex spaces.

scholarly journals Continuity of Convolution of Test Functions on Lie Groups

2014 ◽  
Vol 66 (1) ◽  
pp. 102-140
Author(s):  
Lidia Birth ◽  
Helge Glöckner
Keyword(s):  
Continuous Functions ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Locally Convex Spaces ◽  
Test Functions ◽  
Bilinear Map ◽  
Locally Compact ◽  
Compactly Supported ◽  
Locally Convex

AbstractFor a Lie group G, we show that the map taking a pair of test functions to their convolution, is continuous if and only if G is σ-compact. More generally, consider with t ≤ r + s, locally convex spaces E1, E2 and a continuous bilinear map b : E1 × E2 → F to a complete locally convex space F. Let be the associated convolution map. The main result is a characterization of those (G; r; s; t; b) for which β is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported L1-functions and convolution of compactly supported Radon measures.

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