Translation of measure algebras and the correspondence to Fourier transforms vanishing at infinity
Let G denote a locally compact (not necessarily abelian) group and M(G) the collection of finite regular Borel measures on G. The set M(G) is a semisimple Banach algebra with identity under convolution *. It can be identified with the dml space of CO(G), the space of continuous complex-valued functi...
| Main Authors: | , |
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| Format: | Text |
| Language: | English |
| Published: |
1970
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| Subjects: | |
| Online Access: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.302 http://www.math.virginia.edu/~der/pdf/der06.pdf |
| Summary: | Let G denote a locally compact (not necessarily abelian) group and M(G) the collection of finite regular Borel measures on G. The set M(G) is a semisimple Banach algebra with identity under convolution *. It can be identified with the dml space of CO(G), the space of continuous complex-valued functions on G that vanish at infinity, with the sup-norm. The group G has a left-invariant regular Borel measure din(x) that is unique up to a constant and is called the left Haar measure of G. Let C ‘(G) denote the space of bounded continuous functions on G. For each x e G, we define on C ‘(G) the left-translation operator by the relation L(x)f(y) = f(x-l y) (f ~ CB(G)). We say that f e CB(G) is right uniformly continuous if L(xa) f % L(x) f uniformly, whenever Xa ~ x. Let C&(G) denote the subspace of C ‘(G) of right uniformly continuous functions. For p c M(G), define L(x) p c M(G) by the condition ~ f(t)dL(x)~(t) = ~ L(x-l)f(t)dw(t), G G where f c CO(G). We wish to study for which p 6 M(G) the map x b L(x) # is continuous from G into M(G), where M(G) will be equipped with an L(x)-invariant metric topology. In particular, we shall characterize MO(G), the algebra of measures whose Fourier transform vanishes at infinity. Let A C C ~U(G) be a linesr subspace with sufficiently many elements to separate the points of M(G); in other words, if w E M(G) and if for all f E A, then p = O. We are J f(t)dp(t) = O G then able to pair A (f, p) = ~ f(t)dp(t) (f c A; G and M(G) by the relation p E M(G)). Let u (A, M(G)) denote the weak topology on A induced by this pairing. Suppose A can be written as UJ.l Ak, where each A ~ is a subset of A that is L(x)-ima.riant for all x 6 G and where each Ak is u (A, M(G))-bounded. Note that Ak is |
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