Isometric isomorphisms between Banach algebras related to locally compact groups. (English) Zbl 0711.43002
It is proved that if T is an isometric isomorphism from the Banach algebra \(LUC(G_ 1)^*\) (the continuous dual of the Banach space of left uniformly continuous functions on \(G_ 1\), equipped with Arens multiplication) onto \(LUC(G_ 2)^*\), then T maps \(M(G_ 1)\) onto \(M(G_ 2)\) and \(L^ 1(G_ 1)\) onto \(L^ 1(G_ 2)\) (consequently \(G_ 1\) and \(G_ 2\) must be isomorphic by Wendel’s theorem). Any isometric isomorphism from \(L^ 1(G_ 1)^{**}\) (second conjugate algebra of \(L^ 1(G_ 1))\) onto \(L^ 1(G_ 2)^{**}\) maps \(L^ 1(G_ 1)\) onto \(L^ 1(G_ 2)\). This answers affirmatively a question raised by F. Ghahramani and A. T. Lau [Bull. Lond. Math. Soc. 20, 342-349 (1988; Zbl 0628.43002)] and was proved there for compact and discrete groups. The case of abelian locally compact groups has been already settled by A. T. Lau and V. Losert [J. Lond. Math. Soc., II. Ser. 37, 464-470 (1988; Zbl 0608.43002)].
Reviewer: H.Rindler
MSC:
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
| 43A10 | Measure algebras on groups, semigroups, etc. |
| 22D15 | Group algebras of locally compact groups |
Keywords:
isometric isomorphism; Banach algebra; continuous dual; Banach space of left uniformly continuous functions; Arens multiplication; locally compact groupsReferences:
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