The spectral topology in rings

Abstract

The spectral topology of a ring is easily defined, has familiar applications in elementary Banach algebra theory, and appears relevant to abstract Fredholm and stable range theory. 0. Introduction. The spectral topology of a ring is motivated by the simple Banach algebra observation that, for x ∈ A, (0.1) ‖x‖ < 1 ⇒ 1− x ∈ A−1. This leads to simple observations about the (norm) closure of the invertible group and its intersection with the one-sided invertibles, relatively regular and Fredholm elements. In a topological algebra the relationship between the spectral and the original topology is able to decide whether or not the invertible group is open, and the spectral topology offers another version of the idea of “stable rank”. Precisely what is the spectral closure of the invertibles in the familiar disc algebra seems to be a delicate question. In this writing each definition and theorem ushers in a new section. We define the spectral closure in §1, prove it gives a topology in §2, with jointly continuous addition and multiplication; in §3 we check some “spectral permanence” and work out some obvious examples. In §4 we make the observation that the spectral topology is the weakest ring topology giving an open group of invertibles, and in §5 we suggest that it offers a possible extension of the “quasinilpotent” concept to general rings. In §6 we chart the interaction between the spectral topology and the “regular” elements of the ring. In §7 we find that onto homomorphisms T : A → B are automatically continuous, that Gelfand homomorphisms are relatively open, and in §8 that if the homomorphism has inverse closed range then the nearly invertible “Fredholm” elements are “Weyl”. In §9 we make an extension of the spectral topology to the space An of n-tuples, and use it to show, in §10, 2010 Mathematics Subject Classification: Primary 46L05.