By Edwin Hewitt, Kenneth A. Ross

Contents: Preliminaries. - components of the speculation of topolo- gical teams. -Integration on in the neighborhood compact areas. - In- variation functionals. - Convolutions and staff representa- tions. Characters and duality of in the community compact Abelian teams. - Appendix: Abelian teams. Topological linear spa- ces. creation to normed algebras. - Bibliography. - In- dex of symbols. - Index of authors and phrases.

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Extra info for Abstract Harmonic Analysis: Volume 1: Structure of Topological Groups Integration Theory Group Representations

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8) are largely due to A. MARKOV [1]. 8) appear in D. VAN DANTZIG [3]. v) known to us is in A. WEIL [3], p. 11. See also VAN DANTZIG [3J, TG. 22. 9) must have been long known; the first reference we have found is a special case proved in MONTGOMERYandZIPPIN [1J,p. 55. 10)isduetovANKAMPEN [1]. Uniform spaces are the creation of A. WEIL [3J, who also described thcuniform structures 9l(G) and Y'r(G). 17) are essen tially due to WEIL [loe. ]. See also VAN KAMPEN [1 J, p. 451. § 5. Subgroups and quotient groups In this section, we take up some methods of forming new topological groups from a given topological group.

If both GJH and H are eompactly generated, then G is also compactly generated. [Let {xH: XEX} be a closed compact subset of GJH that generates GJH.

We now prove (iv). Suppose that ab=ba for all aEA, bEB. The mapping (a, b)--+aba-1b-1 of GxG into G is continuous. 8), we infer that H ={(a, b)EGxG:aba-1b-1=e} is closed. Clearly AxBcHand (AxB)-=A-xB-. Thus, wehave A-xB-c H. That is, ab=ba for all aEA-, bEB-. 3) Corollary. 11 H is a subsemigroup, subgroup, or normal subgroup 01 a topologieal group G, then H- is also a subsemigroup, subgroup, or normal subgroup, respeetively, 01 G. 11 G is a To topologieal group and H is an Abelian subsemigroup or subgroup 01 G, then H- is also an Abelian subsemigroup or subgroup, respectively, ot G.

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