Measurability Properties of Spectra

S. Levi, Z. Slodkowski
1986 Proceedings of the American Mathematical Society  
We study Borel measurability of the spectrum in topological algebras. We give some equivalences of the various properties, show that the spectrum in a Banach algebra is continuous on a dense Gs, and prove that in a Polish algebra the set of invertible elements is an FaS and the inverse mapping is a Borel function of the second class. This article has its origin in the papers [7] and [5] . We study Borel measurability of the spectrum and related sets and mappings in various classes of algebras.
more » ... asses of algebras. The best-known example is the case of Banach algebras: the spectrum is then an upper semicontinuous mapping, the set of invertible elements open, and the inverse mapping continuous. In the first part of the paper we establish relations between various measurability properties and study the set of points of measurability of the spectrum considered as a set-valued function. (This part of the paper-in particular Lemma 6 through Corollary 9-can be read independently.) We prove, among other things, the following fact. Theorem (cf. Theorem 7). If X is a Banach algebra, then the spectrum x -> o(x): X -» 2C is continuous on a dense Gs in X. The second part of the paper uses topological methods to deduce measurability results: we prove that in a Polish algebra the set of invertible elements is an FaS and the inverse mapping is a Borel measurable function of the second class. We prove in particular (Theorem 13), without any assumption of separability or local convexity, that the spectrum of a continuous linear operator acting on a complete metrizable vector space is always a GSa set. This answers a question posed by A. L. Shields [6]. Terminology. Throughout the paper X will denote a complex algebra with identity e (commutativity is not assumed), which is also a topological vector space. The continuity properties of multiplication will be specified case by case. An F-space is a complete metric vector space. A Polish space is a complete separable metric space. A topological space Z is said to be a Baire space if every nonempty open subset of Z is of second category (i.e. cannot be represented as the union of a countable family of nowhere dense subsets of Z).
doi:10.2307/2045689 fatcat:de6mf7uzerg2zoysbsdmvm67xq