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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homomorphisms on groups and induced maps on certain algebras of measures


Authors: Charles F. Dunkl and Donald E. Ramirez
Journal: Trans. Amer. Math. Soc. 160 (1971), 475-485
MSC: Primary 22.20; Secondary 42.00
MathSciNet review: 0283129
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Abstract: Suppose that $ \varphi $ is a continuous homomorphism of a locally compact group $ G$ into another such group, $ H$, then $ \varphi $ induces in a natural way a homomorphism $ {\varphi ^ \ast }$ of the measure algebra of $ G$, called $ M(G)$, into $ M(H)$. The action of $ {\varphi ^ \ast }$ on the subspace $ {M_0}(G)$ is studied in this paper. The space $ {M_0}(G)$ is the nonabelian analogue to the space of measures on a locally compact abelian group whose Fourier-Stieltjes transforms vanish at infinity, and is defined herein. We prove that if $ \varphi $ is an open homomorphism then $ {\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H)$. If $ G$ and $ H$ are abelian and $ \varphi $ is not open, then $ {\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\} $. The main tool for this theorem is the fact, proved herein, that $ \varphi $ is open if and only if its adjoint, $ \hat \varphi :\hat H \to \hat G$, is proper (where $ \hat G,\hat H$ are the character groups of $ G,H$ resp.). Further properties of $ {M_0}(G)$ for abelian or compact groups $ G$ are derived.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0283129-7
PII: S 0002-9947(1971)0283129-7
Keywords: Proper homomorphism, open homomorphism, measure algebra, translation of measures is continuous, translation invariant norm on measures, Fourier-Stieltjes transforms, Fourier-Stieltjes transforms vanishing at infinity, local direct product of groups, ergodic homomorphism, band of measures
Article copyright: © Copyright 1971 American Mathematical Society