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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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