A note on the almost left and almost right joint spectra of R. Harte. (English) Zbl 0704.46028

Sołtysiak, Andrzej

Commentat. Math. Univ. Carol. 30, No. 2, 317-320 (1989).

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

Keywords:

complex unitial normed algebra; nonzero continuous multiplicative linear functional; almost left (or right) joint spectrum