scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

Counterexample to a Conjecture of Greenleaf

1970 ◽  
Vol 13 (4) ◽  
pp. 497-499 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Image Position

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

Bipositive and Isometric Isomorphisms of Some Convolution Algebras

1965 ◽  
Vol 17 ◽  
pp. 839-846 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Lebesgue Space ◽  
Inductive Limit ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Natural Topology ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Image Position ◽  
Complex Valued

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

On a question of R. Godement about the spectrum of positive, positive definite functions

1991 ◽  
Vol 110 (1) ◽  
pp. 137-142
Author(s):  
Mohammed B. Bekka
Keyword(s):  
Compact Group ◽  
Uniform Convergence ◽  
Unitary Representation ◽  
Convex Set ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Image Position ◽  
Compact Subsets

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

scholarly journals Some examples of nonmeasurable sets

1974 ◽  
Vol 18 (2) ◽  
pp. 236-238 ◽  
Author(s):  
Edwin Hewitt ◽  
Karl Stromberg
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Outer Measure ◽  
Recent Issue ◽  
Locally Compact ◽  
Image Position ◽  
Nonmeasurable Sets

In a recent issue of this Journal, Pu [3] has given an interesting construction of a nonmeasurable subset A of R such that for all intervals I in R. [Throughout this note, the symbol λ denotes Lebesgue outer measure on R or Haar outer measure on a general locally compact group.] This solves a problem stated in [2], p. 295, Exercise (18.30).

scholarly journals A Duality of Locally Compact Groups That Does Not Involve the Haar Measure

2015 ◽  
Vol 116 (2) ◽  
pp. 250 ◽  
Author(s):  
Yulia Kuznetsova
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
C Algebras ◽  
Commutative Algebras

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf $C^*$-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the $C^*$-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

Convolutions as Bilinear and Linear Operators

1964 ◽  
Vol 16 ◽  
pp. 275-285 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Linear Operators ◽  
Locally Compact ◽  
Radon Measures

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

S.I.P. measures on metrizable groups

1983 ◽  
Vol 93 (3) ◽  
pp. 511-518
Author(s):  
C. Karanikas
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Convolution Powers ◽  
Image Position ◽  
The Right

Let G be a metrizable non-discrete locally compact group. Let M(G) be the convolution measure algebra of G. We shall denote by Lx and Rx, x β G, the left and the right translation operation on M(G), respectively. A measure μ β M(G) has strongly independent powers or μ is a s.i.p. measure, if, for x β G and n, m β N,whenever n ≠ m or x ≠ e, where e is the identity of G. If a measure μ satisfies (1) for x = e and n + m, we say that μ has independent powers, or μ is an i.p. measure. Notice that the powers μn and μm of μ are convolution powers.

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