# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Schur Lemma and the Spectral Mapping Formula

### Tom 55 / 2007

Bulletin Polish Acad. Sci. Math. 55 (2007), 63-69 MSC: Primary 46H10; Secondary 46H15, 46H30. DOI: 10.4064/ba55-1-7

#### Streszczenie

Let $B$ be a complex topological unital algebra. The left joint spectrum of a set $S\subset B$ is defined by the formula $$\sigma_l(S)=\{(\lambda(s))_{s\in S}\in\mathbb C^S\mid \{s-\lambda(s)\}_{s\in S} \hbox{ generates a proper left ideal}\}.$$ Using the Schur lemma and the Gelfand–Mazur theorem we prove that $\sigma_l(S)$ has the spectral mapping property for sets $S$ of pairwise commuting elements if

(i) $B$ is an m-convex algebra with all maximal left ideals closed, or

(ii) $B$ is a locally convex Waelbroeck algebra.

The right ideal version of this result is also valid.

#### Autorzy

• Antoni WawrzyńczykDepartamento de Matemáticas