Local uniform and uniform convexity of non-commutative symmetric spaces of measurable operators

Abstract

Let E be a separable symmetric sequence space, and let CE be the unitary matrix space associated with E, i.e. the Banach space of all compact operators x on l2 so that s(x) E, with the norm , where are the s-numbers of x. One of the interesting subjects in the theory of the unitary matrix spaces is the clarification of correlation between the geometric properties of the spaces E and CE. A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces CE has been already obtained (see 13).