Abstract
We show that if (E, ∥ · ∥E) is a symmetric Banach sequence space then the corresponding space 
of operators on a separable Hilbert space, defined by 
if and only if 
, is a Banach space under the norm 
. Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that 
is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.
© Walter de Gruyter Berlin · New York 2008
