The complex convexity of quasi-normed linear spaces

Abstract

A notion of uniform convexity is defined for quasi-normed (complex) spaces by replacing norms of midpoints of segments in the space by norms of centers of complex discs in the space. Complex uniformly convex spaces (c.u.c. spaces) always have cotype. In Banach lattices, possessing cotype and being complex uniformly convex are the same. Duals of C ∗-algebras are c.u.c., and, in fact, have cotype 2. The behavior of certain martingale difference sequences in c.u.c. spaces is examined. This leads to an isomorphic description of the spaces having the property.