Weak approximate identities and multipliers
Weak invertibility and strong spectrum
A notion of weak invertibility in a unital associative algebra A and a corresponding notion of strong spectrum of an element of A is defined. It is shown that many relationships between the Jacobson radical, the group of invertibles and the spectrum have analogues relating the strong radical, the set of weakly invertible elements and the strong spectrum. The nonunital case is also discussed. A characterization is given of all (submultiplicative) norms on A in which every modular maximal ideal M...
When a unital F-algebra has all maximal left (right) ideals closed?
We prove that a real or complex unital F-algebra has all maximal left ideals closed if and only if the set of all its invertible elements is open. Consequently, such an algebra also automatically has all maximal right ideals closed.
When unit groups of continuous inverse algebras are regular Lie groups
It is a basic fact in infinite-dimensional Lie theory that the unit group of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group is regular in Milnor’s sense. Notably, is regular if A is Mackey-complete and locally m-convex.
Where to find the image of a derivation
With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.