Three spectral notions for representations of commutative Banach algebras

Abstract

Let T be a bounded representation of a commutative Banach algebra B. The following spectral sets are studied. Λ 1 (T): the Gelfand space of the quotient algebra B/ Ker T. Λ 2 (T): the Gelfand space of the operator algebra Im T. Λ 3 (T): those characters ϕ of B for which the inequalities ∥T b x-b ^(ϕ)x∥<ε∥x∥, b∈F, have a common solution x≠0, for any ε>0 and any finite subset F of B. A theorem of Beurling on the spectrum of L ∞ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for T the regular representation and its adjoint.