Uniform crossnorms

Barry Simon

Vol. 46 (1973), No. 2, 555–560

DOI: 10.2140/pjm.1973.46.555

Abstract

A crossnorm on a pair of Banach spaces (X,Y ) is a norm, α, on the algebraic tensor product X ⊙ Y obeying α(x ⊗ y) = ∥x∥∥y∥ for alI x ∈ X,y ∈ Y. When Schatten introduced crossnorms, he singled out two general classes of crossnorms: the dualizable crossnorms (called by him “crossnorms whose associates are crossnorms”) and the uniform crossnorms. These are crossnorms which induce in a natural way other crossnorms: in the dualizable case, a crossnorm, α_d, on X^∗⊙ Y ^∗, and in the uniform case, a crossnorm, α, on ℒ(X) ⊙ℒ(Y ) where ℒ(X) is the algebra of bounded operators on X. Our main new result is a proof that if α is a uniform crossnorm, then α, the induced crossnorm on ℒ(X) ⊙ℒ(Y ) is dualizable.

Mathematical Subject Classification 2000

Primary: 46M05

Milestones

Published: 1 June 1973