An approach to numerical ranges without Banach algebra theory

Lie algebra theory without algebra

This is an entirely expository piece: the main results discussed are very wellknown and the approach we take is not really new, although the presentation may be somewhat different to what is in the literature. The author’s main motivation for writing this piece comes from a feeling that the ideas deserve to be more widely known. Let g be a Lie algebra over R or C. . A vector subspace I ⊂ g is a...

Unimodular Matrices in Banach Algebra Theory

Let A be a ring with 1 and denote by L (resp. R) the set of left (resp. right) invertible elements of A. If A has an involution *, there is a natural bijection between L and R. In general, it seems that there is no such bijection; if A is a Banach algebra, L and R are open subsets of A, and they have the same cardinality. More generally, we prove that the spaces Uk(A") of n X i-left-invertible ...

Math 206: Banach Algebra and Spectral Theory

An amenable, radical Banach algebra

We give an example of an amenable, radical Banach algebra, relying on results from non-abelian harmonic analysis due to H. Leptin, D. Poguntke and J. Boidol. Let A be a Banach algebra, and let E be a Banach A-module. A bounded linear map D : A → E is called a a derivation if D(ab) = a.Db+ (Da).b (a, b ∈ A). A derivation D : A → E is said to be inner if Da = x.a− a.x (a ∈ A) for some x ∈ E. For ...

Square Roots and Powers in Constructive Banach Algebra Theory

Several new and improved results about positive integral powers of hermitian elements, and square roots of positive elements, in a Banach algebra are proved constructively.