Journal ArticleDOI
On homomorphisms of group algebras.
TLDR
In this article, it has been shown that a Fourier-Stieltjes transform taking only the values zero and one must either be identically zero or identically one.Abstract:
Introduction. There have been several papers written on the subject of determining all homomorphisms, and more particularly, isomorphisms of group algebras of locally compact abelian groups. Except in certain special cases there does not seem to be much known. One case which has been treated is the case of two groups G and H, where H has a connected dual group L . In this case it has been shown that the only homomorphisms of L1(G) into L1(H) are essentially those induced by homomorphisms of G into H [1]. This result was proved in the case where H is the real line and by means of the structure theory of locally compact abelian groups, extended to the more general case. The crucial point in the proof seems to be the obvious fact that a Fourier-Stieltjes transform taking only the values zero and one must either be identically zero or identically one. Equivalently we may say that the only idempotent measures on H are the zero measure, and Haar measure of the identity subgroup. Another case, that in which H is the circle group, has been solved in [5], [6]. In this case too, the complete analysis of -idempotent measures on the circle achieved in [4], was very heavily used. The author in a previous paper [2], has determined all the idempotent measures on locally compact groups. Thus, it seems reasonable that one should now be able to completely solve the homomorphism problem. In this paper, we shall do exactly that. A very simple, but hitherto unnoticed, relationship between the homomorphism problem and idempotent measures is established in the case of compact G and H. Then a passage to the Bohr compactifications of the groups in question yields the general result. There are certain technical complications which appear, some of which are standard, such as convolutions with approximate identities, which we hope will not confuse the reader. It is perhaps unnecessary to add that at all times the reader should bear in mind the concrete examples of Fourier series and Fourier integrals to better understand what is happening. In the case of Fourier series, our problem is precisely one of determining which mappings of Fourier coefficients in m-variables into coefficients in n-variables, send Fourier coefficients into Fourier coefficients. More specifically, let 7r be aread more
Citations
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Journal ArticleDOI
Completely bounded homomorphisms of the Fourier algebras
Monica Ilie,Nico Spronk +1 more
TL;DR: For locally compact groups G and H, this article showed that if G is amenable then any completely bounded homomorphism Φ : A (G ) → B (H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group.
Journal ArticleDOI
Norm decreasing homomorphisms of group algebras
Journal ArticleDOI
Amenability and ideals in a(g)
TL;DR: Closed ideals in A(G) with bounded approximate identities are characterized for amenable [SIN]-groups and arbitrary discrete groups in this article, which extends a result of Liu, van Rooij and Wang for abelian groups.
Journal ArticleDOI
Bimeasure algebras on LCA groups.
TL;DR: In this paper, a multiplication and an adjoint operation are introduced on BM{Gλ9 G2] which generalize the convolution structure of M(G X H) and which make BM(GU G2) into a Λ^-Banach *-algebra, where KG is Grothendieck's universal constant.
Journal ArticleDOI
On the coset ring and strong Ditkin sets
TL;DR: In this article, the authors presented a complete description of the closed sets in the coset ring of an abelian topological group G and showed that every such set in a separable, metrizable, locally compact, topology group Γ is a strong Ditkin set in the sense of Wik.
References
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Journal ArticleDOI
On a Conjecture of Littlewood and Idempotent Measures
TL;DR: In this article, the problem of determining all the idempotent measures on a locally compact abelian group is studied. But the problem is not restricted to the circle group; it is also applicable to the more general class of groups.
Journal ArticleDOI
Fourier-Stieltjes transforms with bounded powers
Henry Helson,Arne Beurling +1 more
Journal ArticleDOI
Isomorphisms of abelian group algebras
TL;DR: Theorem 4 as mentioned in this paper is a modification of a proof of A. Wendel's theorem for non-abelian group algebras, which is based on the Fourier transform.
Journal ArticleDOI
On isomorphisms of group algebras
TL;DR: In this paper, it was shown that the group algebra of the circle isomorphic to that of the torus, and the theorem announced here stems from this question, which has been left unanswered.