Real Gel’fand-mazur Division Algebras

We show that the complexification (Ã, τ̃) of a real locally pseudoconvex (locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra (A,τ) is a complex locally pseudoconvex (resp., locally absorbingly pseudoconvex, locally multiplicatively pseudoconvex, and exponentially galbed) algebra and all elements in the complexification (Ã, τ̃) of a commutative real exponentially galbed algebra (A,τ)with bounded elements are bounded if the multiplication in (A,τ) is jointly continuous. We give conditions for a commutative strictly real topological division algebra to be a commutative real Gel’fand-Mazur division algebra.