## Regular spectrum of elements in topological algebras.(English)Zbl 1410.46028

Let $$A$$ be a topological algebra (with separately continuous multiplication). An element $$a \in A$$ is said to be bounded if there is a positive number $$\lambda$$ such that the sequence $$\{ (\frac{a}{\lambda})^n \}_n$$ is bounded in $$A$$. We denote by $$A_0$$ the set of bounded elements of $$A$$ and by $$\operatorname{Qinv}A$$ the set of quasi-invertible elements of $$A$$.
In [J. Math. Pures Appl. (9) 33, 147–186 (1954; Zbl 0056.33601)], L. Waelbroeck introduced the regular spectrum for unital commutative locally convex algebras and in [Proc. Lond. Math. Soc. (3) 15, 399–421 (1965; Zbl 0138.38202)], G. R. Allan defined the regular spectrum for general unital locally convex algebras. In the paper under review, the author generalizes Allan’s definition to the case of a general topological algebra $$A$$, which is not necessarily unital. Let $$a \in A$$. The regular spectrum $$\mathrm{sp}_A^r (a)$$ is $\mathrm{sp}_A^r (a)= \mathrm{sp}_A(a) \cup \left\{\lambda \in \mathbb C \setminus \{0\}: \frac{a}{\lambda} \in \operatorname{Qinv}A \text{ and } \left(\frac{a}{\lambda}\right)^{-1}_q \not\in A_0\right\} \cup S ,$ where $$S= \{ \infty\}$$ if $$a \not\in A_0$$ and $$S= \emptyset$$ otherwise. The paper is devoted to the study of properties of the regular spectrum. In particular, the author discusses properties of the disolvent map $$D_a$$, for $$a \in A$$ in the case where $$A$$ is a non-unital topological algebra, with $D_a : \mathbb C\setminus \mathrm{sp}_A(a) \to \mathbb C, \quad \lambda \mapsto \left(\frac{a}{\lambda}\right)^{-1}_q.$

### MSC:

 46H05 General theory of topological algebras 46H20 Structure, classification of topological algebras

### Citations:

Zbl 0056.33601; Zbl 0138.38202
Full Text:

### References:

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