Renormings of c_0(Γ)

A biorthogonal system in a Banach space is called Auerbach whenever both the vectors and the associated functionals are precisely of norm 1. We will show that assuming the Continuum Hypothesis, there exist renormings of c_0(\omega_1) that do not contain uncountable Auerbach systems, which contrasts with another result asserting that big enough Banach spaces always contain uncountable Auerbach systems. As a consequence we conclude that c_0(\omega_1) with such renorming does not embed isometrically into spaces of bounded functions with countable supports.
Based on joint work with P. Hajek and T. Russo as well as with W. B. Johnson.