Two generalizations of the Gleason–Kahane–Zelazko theorem

Erik Christensen

Vol. 177 (1997), No. 1, 27–32

DOI: 10.2140/pjm.1997.177.27

Abstract

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Zelazko Theorem. We consider a unital Banach algebra A, and a continuous unital linear mapping φ of A into M_n(ℂ) – the n × n matrices over ℂ. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for φ in saying that φ(A_inv) is contained in GL_n(ℂ) if and only if for each a in A and each natural number k:

Milestones

Received: 19 April 1995

Revised: 18 September 1995

Published: 1 January 1997