Compact left multipliers on Banach algebras related to locally compact groups. (English) Zbl 1169.43001
Given a locally compact group \(G\), let \(L^\infty_0(G)\) be the Banach algebra of equivalence classes of essentially bounded functions vanishing at infinity, and \(A:=L^\infty_0(G)^*\) be the dual Banach algebra, equipped with the Arens product [see A. T.-M. Lau and J. Pym, J. Lond. Math. Soc., II. Ser. 41, No. 3, 445–460 (1990; Zbl 0667.43004)]. A continuous linear map \(T: A\to A\) is called a left multiplier if \(T(ab)=T(a)b\) for all \(a,b\in A\). A left multiplier is called compact if it is a compact operator. As the main result, the authors show that \(A\) admits a compact left multiplier if and only if \(G\) is compact. This complements studies of multipliers on the second dual algebras \(L^1(G)^{**}\) and \(M(G)^{**}\) by F. Ghahramani and A. T.-M. Lau (see [Math. Proc. Camb. Philos. Soc. 111, No. 1, 161–168 (1992; Zbl 0818.46050)] and subsequent work).
Reviewer: Helge Glöckner (Paderborn)
MSC:
43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
46H05 | General theory of topological algebras |
47B07 | Linear operators defined by compactness properties |
47B48 | Linear operators on Banach algebras |
Keywords:
Arens product; compact operator; multiplier; completely continuous element; locally compact group; compactnessReferences:
[1] | DOI: 10.1112/jlms/s2-41.3.445 · Zbl 0667.43004 · doi:10.1112/jlms/s2-41.3.445 |
[2] | DOI: 10.1112/jlms/s2-35.1.135 · Zbl 0585.43001 · doi:10.1112/jlms/s2-35.1.135 |
[3] | Hewitt, Real and Abstract Analysis (1975) |
[4] | Hewitt, Abstract Harmonic Analysis I (1970) |
[5] | DOI: 10.2307/2001602 · Zbl 0711.43002 · doi:10.2307/2001602 |
[6] | DOI: 10.1006/jfan.1997.3133 · Zbl 0891.22007 · doi:10.1006/jfan.1997.3133 |
[7] | DOI: 10.1090/S0002-9947-96-01499-7 · Zbl 0859.43001 · doi:10.1090/S0002-9947-96-01499-7 |
[8] | DOI: 10.1017/S0305004100075241 · Zbl 0818.46050 · doi:10.1017/S0305004100075241 |
[9] | Akemann, Pacific J. Math. 22 pp 1– (1967) · Zbl 0158.14205 · doi:10.2140/pjm.1967.22.1 |
[10] | Sakai, Pacific J. Math. 14 pp 659– (1964) · Zbl 0135.35803 · doi:10.2140/pjm.1964.14.659 |
[11] | Mehdipour, Bull. Austral. Math. Soc. 76 pp 49– (2007) |
[12] | DOI: 10.1016/j.jfa.2003.10.012 · Zbl 1069.43001 · doi:10.1016/j.jfa.2003.10.012 |
[13] | DOI: 10.1006/jfan.1995.1104 · Zbl 0832.22007 · doi:10.1006/jfan.1995.1104 |
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