Duality and harmonic analysis on central topological groups. II

✍ Scribed by Siegfried Grosser; Richard Mosak; Martin Moskowitz

• Publisher: Elsevier Science
• Year: 1973
• Weight: 851 KB
• Volume: 76
• Category: Article
• ISSN: 1385-7258
• DOI: 10.1016/1385-7258(73)90040-1

✦ Synopsis

In [8] a Fourier inversion formula was derived for functions in the linear span [8(Li) n @] of jj(L1) n @, where $J(Li) is the center of L,(G) and @ is the set of continuous positive-definite functions on G. For such functions f the function e I+ &f*(e) is in Li(G^, dp), and one has (*I f(x) = SG~ f^(e) x&4-44e) (x E G) where f^(e) is the trace of the Fourier transform F~(Q), and CI$ is the Plancherel measure on CT* given in [8, section 41. We now give a general inversion theorem valid for functions which are not necessarily central. First we prove the following preliminary form of the inversion theorem directly generalizing the inversion theorem of [8]. (3.1) PROPOSITION: Let G be a [Q-group, and f E [Li n @I. Then for all z E G, the function Q I+ u$tr(!Z'f(e)e(z)*) is in Li(G^, dp), and one has for all z E G, PROOF: We observe first that when f is a central Li-function, then WA=~~lf"W, h ence c&tr(T&) Q(X)*) = f^(e) x&r)-. This shows that the formula (**) reduces to (*) if f E &Cl). Now, if f E [Li n @I, then also g = f# E [L1 n @>I, where f#(x) = JG,Z f(ms-1) 0% (see [lo, (l.l)] and [19, (1.3)]). Moreover, gE B&(G)), g(l)=f(l), and g^(e)=f^(e) for all e E GA [8, (6.7)]. I n view of the inversion theorem for central functions [8, (S.ll)], this shows that the function Q I-+ &f-(e) is

integrable on GA. Also, applying the inversion formula (*) to g, evaluating at x = 1, and using the relationships between g and f, we get