scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

Spectra of independent power measures

1972 ◽  
Vol 72 (1) ◽  
pp. 27-35 ◽  
Author(s):  
W. J. Bailey ◽  
G. Brown ◽  
W. Moran
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Elementary Proof ◽  
Natural Generalization ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Power Measure ◽  
Lca Groups

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

On the Commutant of Certain Automorphism Groups

1973 ◽  
Vol 25 (6) ◽  
pp. 1165-1169 ◽  
Author(s):  
P. K. Tam
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Automorphism Groups ◽  
Locally Compact Abelian Group ◽  
Dense Subgroup ◽  
Locally Compact

Let be a W*-algebra, A ( ) the group of all automorphisms of . In this paper we have determined the commutant G’ of a subgroup G of A () for certain classes of G and . The main results are as follows.Theorem 1. If G is a locally compact abelian group acting by translation on the W*-algebra L∞(G), then the commutant of a dense subgroup of G is G itself.

scholarly journals A theorem on abstract Segal algebras over some commutative Banach algebras

1982 ◽  
Vol 25 (2) ◽  
pp. 293-301 ◽  
Author(s):  
U.B. Tewari ◽  
K. Parthasarathy
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Locally Compact ◽  
Commutative Banach Algebras

Let B be a commutative, semi-simple, regular, Tauberian Banach algebra with noncompact maximal ideal space Δ(B). Suppose B has the property that there is a constant C such that for every compact subset K of Δ(B) there exists a f ∈ B with = 1 on K, ‖f‖B ≤ C and has compact support. We prove that if A is a proper abstract Segal algebra over B then for every positive integer n there exists f ∈ B such that fn ∉ A but fn+1 ∈ A. As a consequence of this result we prove that if G is a nondiscrete locally compact abelian group, μ a positive unbounded Radon measure on Γ (the dual group of G), 1 ≤ p < q < ∞ and , then .

scholarly journals The algebra of functions with Fourier transforms in a given function space

1973 ◽  
Vol 9 (1) ◽  
pp. 73-82 ◽  
Author(s):  
U.B. Tewari ◽  
A.K. Gupta
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Function Space ◽  
Compact Abelian Group ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Algebra Of Functions

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

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