Characterization of measures in the group \(C^*\)-algebra of a locally compact group. (English) Zbl 0702.43003
Using P. Eymard’s theory of Fourier and Fourier-Stieltjes algebras on locally compact groups it is shown that measures are in the group \(C^*\)-algebra iff they are in \(C^*_{\rho}\) (the \(C^*\)-algebra corresponding to the reduced dual) and elements in this space \(M_ 0\) are characterized by vanishing on a class \({\mathcal W}\) of Borel subsets of G. This extends earlier results of R. Lyons [Ann. Math., II. Ser. 122, 155-170 (1985; Zbl 0583.43006)] and the author [Math. Ann. 284, 55- 62 (1989; Zbl 0645.43005)] for l.c.a. respectively compact groups.
The proof uses the existence of a.e. Cesàro convergent subsequences of a bounded sequence in \(L^ 2(G)\) first used by Lyons for the abelian case and a characterization of \(M_ 0\) due to C. Dunkl and D. Ramirez as measures on which left translation acts continuously when \(M_ 0\) is endowed with the weak topology induced by the Fourier algebra of G. Sets in \({\mathcal W}\) are defined as Borel subsets of G on which the Cesàro mean of a sequence of functions in the Fourier algebra, which satisfy certain continuity conditions, exists a.e. and is nonzero.
It is shown that on Lie groups \(M_ 0\) is properly contained in the space of continuous measures and contains the space of absolutely continuous measures as a proper subspace. An example of a measure with Fourier-Stieltjes transform vanishing at infinity with respect to the Fell topology which is not in \(M_ 0\) is given on the \(a\cdot x+b\) group.
The proof uses the existence of a.e. Cesàro convergent subsequences of a bounded sequence in \(L^ 2(G)\) first used by Lyons for the abelian case and a characterization of \(M_ 0\) due to C. Dunkl and D. Ramirez as measures on which left translation acts continuously when \(M_ 0\) is endowed with the weak topology induced by the Fourier algebra of G. Sets in \({\mathcal W}\) are defined as Borel subsets of G on which the Cesàro mean of a sequence of functions in the Fourier algebra, which satisfy certain continuity conditions, exists a.e. and is nonzero.
It is shown that on Lie groups \(M_ 0\) is properly contained in the space of continuous measures and contains the space of absolutely continuous measures as a proper subspace. An example of a measure with Fourier-Stieltjes transform vanishing at infinity with respect to the Fell topology which is not in \(M_ 0\) is given on the \(a\cdot x+b\) group.
Reviewer: M.Blümlinger
MSC:
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
| 43A05 | Measures on groups and semigroups, etc. |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
Keywords:
Fourier-Stieltjes algebras; locally compact groups; group \(C^ *\)- algebra; Lie groups; continuous measures; Fourier-Stieltjes transformReferences:
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