B will always denote a
commutative semi-simple Banach algebra with a unit element. If f ∈ B then r(f)
denotes its spectral radius. A sequence F = (fj)1∞ is called a spectral null sequence
if ∥fj∥≦ 1 for each j, while limj→∞r(fj) = 0. If F = (fj) is a spectral null sequence
we put rN(F) =limsupj→∞∥fJN∥1∕N for each N ≧ 1. Finally we define the complex
number rN(B) =sup is a spectral null sequence in . In general
rN(B) = 1 for all N ≧ 1 and the aim of this paper is to study the case when
rN(B) < 1 for some N.
is a spectral null sequence in 
. In general
