Decomposition of 𝐵(𝐺)

Miao, Tianxuan

doi:10.1090/S0002-9947-99-02328-4

Decomposition of $B(G)$
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by Tianxuan Miao PDF

Trans. Amer. Math. Soc. 351 (1999), 4675-4692 Request permission

Abstract:

For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $B^{s}(G)$, where $B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $K\subset G$, there is an $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $K\subset G$ such that $\vert \langle f, b \rangle \vert < \epsilon$ for all $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert$ for all $a\in A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \{ \Vert a b\Vert : a\in A(G), \; supp(a) \subseteq K^{c} \; \text { and } \; \Vert a\Vert \le 1 \}$.

References

Gilbert Arsac, Sur l’espace de Banach engendré par les coefficients d’une représentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), no. 2, 1–101 (French). MR 444833
Larry Baggett and Keith Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), no. 3, 593–600. MR 509261, DOI 10.1090/S0002-9939-1978-0509261-X
L. J. Bunce, The Dunford-Pettis property in the predual of a von Neumann algebra, Proc. Amer. Math. Soc. 116 (1992), no. 1, 99–100. MR 1091177, DOI 10.1090/S0002-9939-1992-1091177-1
Michael Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann. 241 (1979), no. 1, 83–96. MR 531153, DOI 10.1007/BF01406711
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) Contemp. Math., vol. 2, Amer. Math. Soc., Providence, R.I., 1980, pp. 15–60. MR 621850
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
Volker Flory, On the Fourier-algebra of a locally compact amenable group, Proc. Amer. Math. Soc. 29 (1971), 603–606. MR 283138, DOI 10.1090/S0002-9939-1971-0283138-3
—, Eine Lebesgue-Zerlegung und funktorielle Eigenschaften der Fourier-Stieltjes Algebra, Inaugural Dissertation, University of Heidelberg (1972).
E. E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and of the measure algebra $M(G)$, Rocky Mountain J. Math. 11 (1981), no. 3, 459–472. MR 722579, DOI 10.1216/RMJ-1981-11-3-459
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Dirk Mittenhuber and Karl-Hermann Neeb, On the exponential function of an invariant Lie semigroup, Sem. Sophus Lie 2 (1992), no. 1, 21–30. MR 1188629
Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), no. 4, 683–741. MR 513952, DOI 10.1216/RMJ-1978-8-4-683
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264
Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
Keith F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983), no. 2, 183–190. MR 690194, DOI 10.1007/BF01455310