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A note on the almost left and almost right joint spectra of R. Harte. (English) Zbl 0704.46028
It is shown here that a complex unitial normed algebra A has a nonzero continuous multiplicative linear functional if and only if for each finite subset \(\{a_ 1,...,a_ n\}\) of A the almost left (or right) joint spectrum \({\tilde \sigma}{}_{\ell}(a_ 1,..,a_ n)\) (or \({\tilde \sigma}{}_ r(a_ 1,..,a_ n))\) is nonempty, in which \({\tilde \sigma}{}_{\ell}(a_ 1,...,a_ n)=\{(\lambda_ 1,...,\lambda_ n)\in {\mathbb{C}}^ n:\) \(1\not\in [\sum^{n}_{i=1}A(a_ i-\lambda_ i)]^-\}\).
Reviewer: T.Husain
MSC:
46H05 General theory of topological algebras
47A10 Spectrum, resolvent
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