Almost multiplicative linear functionals and approximate spectrum

Document Type : Research Paper

Author

Department of Mathematics, University of Neyshabur

Abstract

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is a linear functional with ϕ(a)∈σ_δ (a) for all a∈A, then ϕ is ε-almost multiplicative. Finally, we use these ideas to provide a sufficient condition for a δ-almost multiplicative functional to be multiplicative.

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Volume 41, Issue 1 - Serial Number 1
February 2015
Pages 177-187
  • Receive Date: 19 December 2012
  • Revise Date: 26 December 2013
  • Accept Date: 27 December 2013
  • First Publish Date: 01 February 2015