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A characterization of homomorphisms in certain Banach involution algebras. (English) Zbl 0756.46024

Let \(A\) be a complex Banach algebra with a symmetric involution such that the space \(\Delta_ A\) of all nontrivial homomorphisms on \(A\) is sigma- compact. The authors show that, for every linear functional \(f\) on \(A\), the following statements are equivalent:
(i) There is a \(k\in\mathbb{C}\setminus\{0\}\) such that \(kf\) is in \(\Delta_ A\).
(ii) For every \(x\) in the kernel of \(f\), there is a homomorphism \(h\) in \(\Delta_ A\) such that \(x\) belongs to the kernel of \(h\).
A commutative version of this result was proved by C. R. Warner and R. Whitley [Proc. Am. Math. Soc., II. Ser. 76, No. 2, 263-267 (1979; Zbl 0497.46034)]. Applications to \(C_ 0(X)\) (where \(X\) is locally compact) and to \(L^ 1(G)\) (where \(G\) is a locally compact metrizable group) are given.

MSC:

46K05 General theory of topological algebras with involution
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46H05 General theory of topological algebras
46H10 Ideals and subalgebras

Citations:

Zbl 0497.46034
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