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On subspectra generated in subalgebras. (English) Zbl 1033.46042
The present paper deals with some properties of ideals in commutative Banach algebras. It is a continuation of the author’s earlier investigations published in [Stud. Math. 142, 245–251 (2000; Zbl 1002.46031) and Bol. Soc. Mat. Mex. (3) 7, 117–121 (2001; Zbl 1041.46035)]. The main result is the following:
Theorem: Let $$A$$ be a unital subalgebra of a commutative Banach algebra $$B$$ with unit $$e$$. If an ideal $$I$$ of $$A$$ consists of non-invertible elements in $$B$$, then there exists an ideal $$J$$ of codimension one in $$A$$ such that $$I\subset J$$ and $$J$$ also consists of non-invertible elements in $$B$$.
Two applications of this result are presented. The first one says that if $$\text{p}=(p_1,\ldots,p_k)$$ is a $$k$$-tuple of polynomials over $$\mathbb C$$ and $$X_{\text{p}}=\{z\in {\mathbb C}^n: p_1(z)=\ldots=p_k(z)\}$$ is an algebraic variety disjoint from a rationally convex compact set $$K$$, then $$X_{\text{p}}$$ can be separated from $$K$$ by a polynomial, i.e., there exists a polynomial $$f$$ such that it is equal to zero on $$X_{\text{p}}$$ and $$0\not\in f(K)$$.
The other one is an application to spectral theory. Namely, from the theorem it follows that the following formula $\tau_B(a)= \Biggl\{(\lambda_1,\ldots,\lambda_k)\in{\mathbb C}^k:\, \sum_{j=1}^kA(a_j-\lambda_je)\text{ is non-invertible in }B\Biggr\},$ where $$a=(a_1,\ldots,a_k)$$ is an arbitrary $$k$$-tuple of elements in $$A$$, defines a certain kind of joint spectrum in the algebra $$A$$. It is shown that $$\tau_B$$ is a subspectrum in the sense of W. Zelazko and a description of it in terms of a certain subset of the maximal ideal space of the algebra $$A$$ is given.
##### MSC:
 46J20 Ideals, maximal ideals, boundaries 46J30 Subalgebras of commutative topological algebras
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