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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