Relative spectra in complete lmc-algebras with applications
Michael E. Boardman · Illinois Journal of Mathematics · 1995
Notation Let e' be a complex (linear associative) algebra with topology induced by a separating directed set of algebra seminorms (11 IIi)i i.Such an algebra is known as a locally multiplicatively-convex (lmc) algebra.We assume through- out that e' is unital and complete in the sense that every net that is Cauchy in each seminorm converges.This definition of lmc-algebra is equivalent to the existence of a local base at 0 consisting of convex idempotent sets [12].However, we are able, in general, to replace the algebra seminorms with equivalent algebra seminorms and still have the same topology on e'.This technique proves to be useful in several examples below.For each I, the set {a z: Ilalli-0} is seen to be an ideal of '.I I [li induces an algebra norm on '/4/.Let / be the completion of the normed algebra ///./ is called the factor algebra associated with II/.We write 7"/-for the canonical homomorphism from " into /.We will make extensive use of the subalgebra of so-called "seminorm-bounded" elements:asg'lllall= supllalli