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joint spectrum; Waelbroeck algebra; commutator; spectral mapping formula
We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $\mathcal A$ to have a non-void left joint spectrum $\sigma _l(S).$ In particular, for a Lie subalgebra $L\subset \mathcal A$ we have $\sigma _l(L)\neq \emptyset $ if and only if $[L,L]$ generates in $\mathcal A$ a proper left ideal. \endgraf 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.
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