On quadratic forms.
We characterize elements in a semisimple Banach algebra which are quasinilpotent equivalent to maximal finite rank elements.
Let a real Banach algebra A with unit be ordered by an algebra cone K. We study the elements a ∈ A with exp(ta) ∈ K, t≥ 0.
A Banach space is called -reflexive if for any cover of by weakly open sets there is a finite subfamily covering some ball of radius 1 centered at a point with . We prove that an infinite-dimensional separable Banach space is -reflexive (-reflexive for some ) if and only if each -net for has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of . We show that the quasireflexive James space is -reflexive for no . We do not know...
We give a spectral characterisation of rank one elements and of the socle of a semisimple Banach algebra.
Without the "scarcity lemma", two kinds of "rank one elements" are identified in semisimple Banach algebras.