The Right Regular Representation of a Compact Right Topological Group

1998 ◽  
Vol 41 (4) ◽  
pp. 463-472 ◽  
Author(s):  
Alan Moran
Keyword(s):  
Topological Group ◽  
Representation Theory ◽  
Regular Representation ◽  
Strong Operator Topology ◽  
Topological Groups ◽  
Compact Topological Group ◽  
Strong Operator ◽  
Counter Example ◽  
The Right ◽  
Compact Right Topological Group

AbstractWe show that for certain compact right topological groups, , the strong operator topology closure of the image of the right regular representation of G in L(H), where H = L2(G), is a compact topological group and introduce a class of representations, R , which effectively transfers the representation theory of over to G. Amongst the groups for which this holds is the class of equicontinuous groups which have been studied by Ruppert in [10].We use familiar examples to illustrate these features of the theory and to provide a counter-example. Finally we remark that every equicontinuous group which is at the same time a Borel group is in fact a topological group.

Representations of Compact Right Topological Groups

1993 ◽  
Vol 36 (3) ◽  
pp. 314-323 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Irreducible Representation ◽  
Regular Representation ◽  
List Type ◽  
Topological Groups ◽  
Regular Representations ◽  
Left Regular Representation ◽  
Left Regular ◽  
Enveloping Semigroups ◽  
The Right

AbstractCompact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centreor a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that: (i)for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.(ii)for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.(iii)when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

Measure in Semigroups

1952 ◽  
Vol 4 ◽  
pp. 396-406 ◽  
Author(s):  
B. R. Gelbaum ◽  
G. K. Kalisch
Keyword(s):  
Topological Group ◽  
Invariant Measure ◽  
Commutative Semigroup ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Counter Example ◽  
Compact Semigroups ◽  
Topological Measures ◽  
Weakened Form ◽  
Locally Compact Semigroups

The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

scholarly journals On topological groups with remainder of character k

2016 ◽  
Vol 17 (1) ◽  
pp. 51
Author(s):  
Maddalena Bonanzinga ◽  
Maria Vittoria Cuzzupè
Keyword(s):  
Topological Group ◽  
Infinite Cardinal ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

<p style="margin: 0px;">In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. <span id="OBJ_PREFIX_DWT1099_com_zimbra_phone" class="Object">42 (2013), 157-163</span>] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k.</p><p style="margin: 0px;"> </p>

Topological aspects of the projective unitary group

1970 ◽  
Vol 68 (1) ◽  
pp. 57-60 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Projective Space ◽  
Unitary Group ◽  
Principal Bundle ◽  
Strong Operator Topology ◽  
Complex Hilbert Space ◽  
Unitary Representations ◽  
Strong Operator ◽  
Unitary Transformations

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

scholarly journals GENERATORS OF REGULAR SEMIGROUPS

2008 ◽  
Vol 50 (1) ◽  
pp. 47-53
Author(s):  
SHMUEL KANTOROVITZ
Keyword(s):  
Regular Semigroup ◽  
Strong Operator Topology ◽  
Holomorphic Extension ◽  
Converse Theorem ◽  
Holomorphic Semigroup ◽  
Regular Semigroups ◽  
Strong Operator ◽  
Basic Facts ◽  
The Right ◽  
Holomorphic Extensions

AbstractA regular semigroup (cf. [4, p. 38]) is a C0-semigroup T(⋅) that has an extension as a holomorphic semigroup W(⋅) in the right halfplane $\Bbb C^+$, such that ||W(⋅)|| is bounded in the ‘unit rectangle’ Q:=(0, 1]× [−1, 1]. The important basic facts about a regular semigroup T(⋅) are: (i) it possesses a boundary groupU(⋅), defined as the limit lims → 0+W(s+i⋅) in the strong operator topology; (ii) U(⋅) is a C0-group, whose generator is iA, where A denotes the generator of T(⋅); and (iii) W(s+it)=T(s)U(t) for all s+it ∈$\Bbb C^+$ (cf. Theorems 17.9.1 and 17.9.2 in [3]). The following converse theorem is proved here. Let A be the generator of a C0-semigroup T(⋅). If iA generates a C0-group, U(⋅), then T(⋅) is a regular semigroup, and its holomorphic extension is given by (iii). This result is related to (but not included in) known results of Engel (cf. Theorem II.4.6 in [2]), Liu [7] and the author [6] for holomorphic extensions into arbitrary sectors, of C0-semigroups that are bounded in every proper subsector. The method of proof is also different from the method used in these references. Criteria for generators of regular semigroups follow as easy corollaries.

scholarly journals Haar measure and compact right topological groups

1992 ◽  
Vol 45 (3) ◽  
pp. 399-413 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Structure Theorem ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Positive Answer ◽  
Topological Groups ◽  
Low Dimensional ◽  
Compact Right Topological Group

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

scholarly journals A-paracompactness and Strongly A-screenability in Topological Groups

2020 ◽  
Vol 13 (2) ◽  
pp. 280-286
Author(s):  
Muhammad Kashif Maqbool ◽  
Awais Yousaf ◽  
Muhammad Siddique Bosan ◽  
Saeid Jafari
Keyword(s):  
Topological Group ◽  
Direct Product ◽  
Open Cover ◽  
Topological Groups ◽  
Compact Topological Group ◽  
Left And Right ◽  
Group Ii

A space is said to be strongly A-screenable if there exists a σ-discrete refinement for each open cover. In this article, we have investigated some of the features of A-paracompact and strongly A-screenable spaces in topological and semi topological groups. We predominantly show that (i) Topological direct product of (countably) A-paracompact topological group and a compact topological group is (countably) A-paracompact topological group. (ii) All the left and right cosets of a strongly A-screenable subset H of a semi topological group (G, ∗, τ ) are strongly A-creenable.

scholarly journals NOTES ON -TOPOLOGICAL GROUPS AND HOMEOMORPHISMS OF TOPOLOGICAL GROUPS

2012 ◽  
Vol 87 (3) ◽  
pp. 493-502 ◽  
Author(s):  
HANFENG WANG ◽  
WEI HE
Keyword(s):  
Topological Group ◽  
Affirmative Answer ◽  
Neutral Element ◽  
Topological Groups ◽  
Open Problems ◽  
Urysohn Space ◽  
Compact Topological Group ◽  
Countable Tightness

AbstractIn this paper, it is shown that there exists a connected topological group which is not homeomorphic to any $\omega $-narrow topological group, and also that there exists a zero-dimensional topological group $G$ with neutral element $e$ such that the subspace $X = G\setminus \{e\}$ is not homeomorphic to any topological group. These two results give negative answers to two open problems in Arhangel’skii and Tkachenko [Topological Groups and Related Structures (Atlantis Press, Amsterdam, 2008)]. We show that if a compact topological group is a $K$-space, then it is metrisable. This result gives an affirmative answer to a question posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl. 104 (2000), 181–190] in the category of topological groups. We also prove that a regular $K$-space $X$ is a weakly Fréchet–Urysohn space if and only if $X$has countable tightness.

Neutrosophic Bi-topological Group

2021 ◽  
pp. 61-67
Author(s):  
Riad K. Al Al-Hamido ◽  
Keyword(s):  
Topological Group ◽  
Continuous Group ◽  
Topological Groups ◽  
Basic Properties ◽  
New Models

Neutrosophic topological groups are neutrosophic groups in an algebraic sense together with neutrosophic continuous group operations. In this article, we have presented neutrosophic bi-topological groups with illustrative examples. We have also defined eight new models of neutrosophic bi-topological groups. Neutrosophic bi-topological group that depends on two neutrosophic topologies group is more general than the neutrosophic topological group. Finally, Some basic properties of neutrosophic bi-topological groups were studied.

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