Uniqueness under spectral variation in the socle of a Banach algebra

Abstract

Let A be a complex semisimple Banach algebra with identity, and denote by σ ′ (x) and ρ(x) the nonzero spectrum and spectral radius of an element x ∈ A, respectively. We explore the relationship between elements a, b ∈ A that satisfy one of the following conditions:The latter problem was identified by Brešar andŠpenko in [7]. In particular, we use these conditions to spectrally characterize prime Banach algebras amongst the class of Banach algebras with nonzero socles, as well as to obtain spectral characterizations of socles which are minimal two-sided ideals.Problem 1. Suppose that a, b ∈ A satisfy σ(ax) = σ(bx) for all x ∈ A. Does this imply a = b?What is the relation between a and b?The first problem has been settled by G. Braatvedt and R. Brits in [5]:2010 Mathematics Subject Classification. 46H05, 46H10, 46H15, 47B10.