An element f of a
commutative Banach algebra is pseudo regular if there is a constant M with
∥abf∥≤ M∥af∥∥bf∥(a,b ∈A). In many cases pseudo regularity implies formally
stronger conditions such as relative invertibility; that is, f is invertible in
some subalgebra of A. In this paper we describe some algebraic methods
which can be used to establish results of this kind. Given a pseudo regular
element f of A, af ∘ bf = abf extends by continuity to a multiplication
∘, called the auxiliary multiplication, on J, the closed ideal generated by
f. This leads to the fundamental inequality ∥ϕ∥J∗≤ M|ϕ(f)| where ϕ is
a multiplicative linear functional on A. As applications of these ideas we
identify the pseudo regular elements of the algebra C(n)[0,1] as being the
elements such that f,f′,…,f(n) have no common zeros and the pseudo regular
elements of the group algebra of a locally compact abelian group as being the
relatively invertible elements. Similar constructions can be made when f
is an element of an A module X though the structure is less rich in this
case.
