scholarly journals Zubov's condition revisited

1983 ◽  
Vol 26 (2) ◽  
pp. 253-257
Author(s):  
Ronald A. Knight
Keyword(s):  
Orbital Stability ◽  
Invariant Set ◽  
Invariant Sets ◽  
Sufficient Condition ◽  
Limit Set ◽  
Necessary And Sufficient Condition ◽  
Considerable Effort ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Compact Boundary

Zubov states an elegant necessary and sufficient limit set condition for positive orbital stability of compact invariant sets in his book “Metody A. M. Lyapunova i ih Primenenie” [11]. Stated in terms of our terminology of L– for the negative limit set, Zubov's proposition is as follows: A necessary and sufficient condition for positive stability of a compact invariant set M isL–(X\M)∩M=Ø. Unfortunately, Zubov's condition L–(X\M)∩M=Ø has subsequently been shown to be necessary but not sufficient (see [9]). Bass and Ura devote considerable effort in [2] and [9[ to correcting Zubov's proposition and Desbrow obtains additional results principally concerning unstable sets in [6] and [7]. Ura gives his classical corrected prolongational version of Zubov's assertion on locally compact phase spaces in [9] and extends it to any closed invariant set with compact boundary on such spaces in [10].

scholarly journals THE MODULUS OF WEAKLY COMPACT MULTIPLIERS ON THE BANACH ALGEBRA L1(𝒢)** OF A LOCALLY COMPACT GROUP

2012 ◽  
Vol 86 (2) ◽  
pp. 315-321
Author(s):  
MOHAMMAD JAVAD MEHDIPOUR
Keyword(s):  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Weakly Compact ◽  
Necessary And Sufficient ◽  
Funct Anal

AbstractIn this paper we give a necessary and sufficient condition under which the answer to the open problem raised by Ghahramani and Lau (‘Multipliers and modulus on Banach algebras related to locally compact groups’, J. Funct. Anal. 150 (1997), 478–497) is positive.

scholarly journals A Riemann integral in a locally compact Hausdorff space

1986 ◽  
Vol 41 (1) ◽  
pp. 115-137 ◽  
Author(s):  
S. I. Ahmed ◽  
W. F. Pfeffer
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Condition ◽  
Lebesgue Integral ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Riemann Integral ◽  
Variational Integrals ◽  
Necessary And Sufficient ◽  
Locally Compact Hausdorff Space

AbstractWe present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

scholarly journals Chaos for Cosine Operator Functions on Groups

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Chung-Chuan Chen
Keyword(s):  
Locally Compact Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Direct Sum ◽  
Cosine Operator ◽  
Weight Condition ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Necessary And Sufficient

Let1≤p<∞andGbe a locally compact group. We characterize chaotic cosine operator functions, generated by weighted translations on the Lebesgue spaceLp(G), in terms of the weight condition. In particular, chaotic cosine operator functions and chaotic weighted translations can only occur simultaneously. We also give a necessary and sufficient condition for the direct sum of a sequence of cosine operator functions to be chaotic.

scholarly journals On orderability of topological groups

1985 ◽  
Vol 8 (4) ◽  
pp. 747-754
Author(s):  
G. Rangan
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Group Structure ◽  
Locally Compact Abelian Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Totally Disconnected

A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

Hecke **-algebras on locally compact hypergroups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
Bentolhoda Sadathoseyni
Keyword(s):  
Compact Group ◽  
Hecke Algebras ◽  
Locally Compact Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Basic Properties ◽  
Necessary And Sufficient

AbstractIn this paper, for a discrete hypergroup K and its subhypergroup H, we initiate the related Hecke {*}-algebra which is an extension of the classical one, and study its basic properties. Especially, we give a necessary and sufficient condition (named (β)) for this algebra to be associative. Also, we show that this new structure is an associative {*}-algebra if and only if K is a locally compact group.

scholarly journals Some properties of distal actions on locally compact groups

2017 ◽  
Vol 39 (5) ◽  
pp. 1340-1360 ◽  
Author(s):  
C. R. E. RAJA ◽  
RIDDHI SHAH
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Contraction Group

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We obtain a decomposition for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.

scholarly journals On the automorphism group of a connected locally compact topological group

1992 ◽  
Vol 35 (2) ◽  
pp. 285-294
Author(s):  
Ta-Sun Wu
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Identity Component ◽  
Necessary And Sufficient ◽  
Locally Compact Topological Group

Let G be a locally compact connected topological group. Let Aut0G be the identity component of the group of all bi-continuous automorphisms of G topologized by Birkhoff topology. We give a necessary and sufficient condition for Aut0G to be locally compact.

On the residual Julia sets of rational functions

1997 ◽  
Vol 17 (1) ◽  
pp. 205-210 ◽  
Author(s):  
SHUNSUKE MOROSAWA
Keyword(s):  
Kleinian Group ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Limit Set ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

Bochner algebras and their compact multipliers

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Fatemeh Abtahi ◽  
Rasoul Isfahani ◽  
Ali Rejali
Keyword(s):  
Radon Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Normed Algebra ◽  
Locally Compact ◽  
Weakly Compact ◽  
Pointwise Multiplication ◽  
Positive Radon Measure ◽  
Necessary And Sufficient

AbstractLet $\mathfrak{A}$ be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L 1(Ω, $\mathfrak{A}$) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L 1(Ω, $\mathfrak{A}$).

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