Mathematica Bohemica, Vol. 143, No. 4, pp. 441-448, 2018

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk

Received September 5, 2017.   Published online June 12, 2018.

Abstract:  We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $\A$ to have a non-void left joint spectrum $\sigma_l(S).$ In particular, for a Lie subalgebra $L\subset\A$ we have $\sigma_l(L)\neq\emptyset$ if and only if $[L,L]$ generates in $\A$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.
Keywords:  joint spectrum; Waelbroeck algebra; commutator; spectral mapping formula
Classification MSC:  47A13, 47A60, 46H30
DOI:  10.21136/MB.2018.0098-17

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Affiliations:   Antoni Wawrzyńczyk, Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, AP 55-534, 09340 México, Ciudad de México, México, e-mail:

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