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

Let E be a separable symmetric sequence space, and let C E be the unitary matrix space associated with E , i.e. the Banach space of all compact operators x on l 2 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 C E . A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces C E has been already obtained (see 13).

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