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The projection property of a family of ideals in subalgebras of Banach algebras. (English) Zbl 1029.46078

Summary: Let \(B\) be unital Banach algebra and \(A\) a unital subalgebra. The family \({\mathcal I}^l_B(A)\) of left ideals of \(A\) consisting of elements non-invertible in \(B\) is studied. We prove that under the condition that \(\alpha Aa^{-1}\subset A\) for all \(a\in A\) which are invertible in \(B\), the family \({\mathcal I}^l_B(A)\) has the following projection property: for every \(k\)-tuple of mutually commuting \(a_1,\dots, a_k\in I\in{\mathcal I}^l_B(A)\) and for every \(c\in A\) commuting with all \(a_i\), there exists \(\lambda\in\mathbb{C}\) and \(J\in {\mathcal I}_B^l(A)\) such that \(a_1,\dots, a_k\), \(c-\lambda e\in J\). It follows that the mapping defined for \(a_1,\dots, a_k\in A\) by the formula \(\sigma_B(a_1, \dots, a_k)= \{(\lambda_1,\dots, \lambda_k)\in \mathbb{C}^k\mid \exists I\in{\mathcal I}_B (A)\), \(a_i-\lambda_ie\in I\), \(1\leq i\leq k\}\) is a subspectrum on \(A\).

MSC:

46J20 Ideals, maximal ideals, boundaries
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