Let A be a Banach *-algebra which is a subalgebra of a Banach algebra B. In this paper, assuming that A is symmetric, various conditions are given which imply that A is inverse closed in B.

A simple application of Pták theory for hermitian Banach algebras, combined with a result on normed Q-algebras, gives a non-technical new proof of the Shirali-Ford theorem. A version of this theorem in the setting of non-normed topological algebras is also provided.

By definition a totally convex algebra $A$ is a totally convex space $\left|A\right|$ equipped with an associative multiplication, i.eȧ morphism $\mu :\left|A\right|\otimes \left|A\right|⟶\left|A\right|$ of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.

Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if $x_1,\dots ,x_n$ and $y_1,\dots ,y_n$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T\left(x_k\right)=y_k$, $k=1,\dots ,n$. We prove that some proper multiplicative subgroups of G have this property.

We generalize Wiener's inversion theorem for Fourier transforms on closed subsets of the dual group of a locally compact abelian group to cosets of ideals in a class of non-commutative *-algebras having specified properties, which are all fulfilled in the case of the group algebra of any locally compact abelian group.