Estimating the Numberal Eigenvalues and Banach – Mazur Rotation Problem with Strong Proximality

Abstract

We extend previous results on operators, in Hilbert space. The method employs complex analysis and a new finite dimensional reduction, allowing us to avoid using the existing theory of determinants in Banach space, which would require strong restriction. We show that the spaces, and all infinite dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. We study product integrability of functions with values in unital Banach algebras. The product integrals are understood in the sense of Kurzweil, Mcshane or Riemann. We investigate a variation of the transitivity problem for proximinality properties of subspace and intersection properties of balls in Banach spaces .