Continuous Dense Embeddings of Strong Moore Algebras

Abstract

Abstract.We introduce the class of strong Moore Banach algebras (all topologically transitive representations of a certain type are finite dimensional) and investigate stability properties with respect to completions of such algebras in continuous submultiplicative norms. Among other things it is shown that every irreducible representation of a regular (definition below) strong Moore Banach algebra j/ extends to all Banach algebras in which stf is continuously and densely embedded.

Topologically Irreducible Representations and Radicals in Banach Algebras

“…Then π = ρφ is a TT representation of A on X. Following Meyer [12], we call such TT representations standard. The intersection of the kernels of all the standard TT representations of a given normed algebra A will be denoted S(A).…”

“…Then every strictly irreducible representation of B induces a continuous topologically irreducible representation of A. This construction is due to Meyer [12], who calls such representations standard. We shall use it in Section 9 to produce a non-commutative Banach algebra in which the radical described above is strictly smaller than the Jacobson radical.…”

Ideal Spaces of Banach Algebras

“…An element p P à A f belongs to the closure of a subset X of à A f in the functional topology if and only if Fp is in the closure of j P X Fj in the weak à -topology. Since there is a canonical one-to-one correspondence between equivalence classes of ®nite-dimensional irreducible representations and their kernels (see [27] for example), we may transfer the functional topology across from à A f to k X k . We shall let t F denote the resulting topology on k X k .…”