The Fourier algebra as an order ideal of the Fourier-Stieltjes algebra. (English) Zbl 0547.43004
For a locally compact group G, let A(G) and B(G) be the Fourier algebra and Fourier-Stieltjes algebra respectively. The authors show the following: A(G) is the smallest nonzero closed order and algebra ideal of B(G).
Reviewer: H.Yamaguchi
MSC:
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 43A40 | Character groups and dual objects |
Keywords:
Fourier algebra; Fourier-Stieltjes algebra; smallest nonzero closed order and algebra idealReferences:
| [1] | Arendt, W., De Cannière, J.: Order isomorphisms of Fourier algebras. J. Funct. Anal.50, 1-7 (1983) · Zbl 0506.43001 · doi:10.1016/0022-1236(83)90057-5 |
| [2] | Arendt, W., De Cannière, J.: Order isomorphisms of Fourier-Stieltjes algebras., Math. Ann.263, 145-156 (1983) · doi:10.1007/BF01456877 |
| [3] | De Cannière, J., Enock, M., Schwartz, J.-M.: Algèbres de Fourier associées à une algèbre de Kac. Math. Ann.245, 1-22 (1979) · doi:10.1007/BF01420426 |
| [4] | De Cannière, J., Enock, M., Schwartz, J.-M.: Sur deux résultats d’analyse harmonique non-commutative: une application de la théorie des algèbres de Kac. J. Operator Theory5, 171-194 (1981) · Zbl 0486.22003 |
| [5] | Dixmier, J.: Les algèbres d’opérateurs dans l’espace hilbertien. Paris: Gauthier-Villars 1969 |
| [6] | Dixmier, J.: LesC *-algèbres et leurs représentations. Paris: Gauthier-Villars 1969 |
| [7] | Effros, E.G.: Order, ideals in aC *-algebra and its dual. Duke Math. J.30, 391-412 (1963) · Zbl 0117.09703 · doi:10.1215/S0012-7094-63-03042-4 |
| [8] | Enock, M., Schwartz, J.-M.: Une dualité, dans les algèbres de von Neumann. Bull Soc. Math. France Suppl. Mém.44, 1-144 (1975) · Zbl 0343.46044 |
| [9] | Ernest, J.: Hopf-von Neumann algebras, pp. 195-215. In: Functional Analysis, Proceedings of a Conference held at the University of California, Irvine (B.R. Gelbaum, ed.), Washington D.C. and London: Thompson Book Company and Academic Press 1967 |
| [10] | Eymard, P.: Algébre de Fourier d’un groupe localement compact. Bull. Soc. Math. France92, 181-236 (1964) · Zbl 0169.46403 |
| [11] | Fell, J.M.G.: Weak containment and induced representations of groups. Canad. J. Math.14, 237-268 (1962) · Zbl 0138.07301 · doi:10.4153/CJM-1962-016-6 |
| [12] | Kirchberg, E.: Darstellungen coinvolutiver Hopf-W *-Algebren und ihre Anwendung in der nicht-Abelschen Dualitätstheorie lokalkompakter Gruppen. Berlin: Akademie der Wissenschaften der DDR 1977 |
| [13] | Pedersen, G.K.:C *-algebras and their automorphism groups. London-New York-San Francisco: Academic Press 1979 · Zbl 0416.46043 |
| [14] | Schaefer, H.H.: Banach lattices and positive operators. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0296.47023 |
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