Harte theorem for Waelbroeck algebras. (English) Zbl 1099.46034

Wawrzyńczyk, Antoni

Math. Proc. R. Ir. Acad. 105A, No. 2, 71-77 (2005).

Summary: Let \(B\) be a locally convex Waelbroeck algebra. Let \(a_1,\dots a_k\in B\) be an arbitrary \(k\)-tuple of mutually commuting elements. The joint spectrum \(\sigma_B(a_1,\dots, a_k)\) is defined as the set of those \((\lambda_1,\dots,\lambda_k)\in \mathbb C^k\) for which the elements \(a_1-\lambda_1,\dots, a_k-\lambda_k\) generate a proper (left or right) ideal. Let \(p:\mathbb C^k\to\mathbb C^m\) be a polynomial mapping. The spectral mapping formula \[ p(\sigma_B(a_1,\dots,a_k))= \sigma_B(p(a_1,\dots, a_k)) \] is proved.

MSC:

46H30	Functional calculus in topological algebras
47A13	Several-variable operator theory (spectral, Fredholm, etc.)