Joint spectra and restricted Gelfand transform. (English) Zbl 07342826
Summary: By considering joint spectrum on infinite and non-commutative subsets of a locally convex Waelbroeck algebra \(W\) we show that the phenomenon called Spectral Mapping Theorem is related with the existence and extensibility of multiplicative functionals on subalgebras generated by subsets \(S\subset W\). The corresponding multiplicative functionals are characterized by the property that \(\ker\varphi\) generates in \(W\) a proper left or right ideal. For a subalgebra \(B\subset W\) which admits this type of functionals we define the restricted Gelfand transform and we provide an interpretation of the related joint spectrum.
MSC:
| 47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |
| 47A60 | Functional calculus for linear operators |
| 46H30 | Functional calculus in topological algebras |
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