Vol. 46, No. 2, 1973

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Uniform crossnorms

Barry Simon

Vol. 46 (1973), No. 2, 555–560

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∥∥yfor 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, αd, on XY , and in the uniform case, a crossnorm, α, on (X) ⊙ℒ(Y ) where (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 (X) ⊙ℒ(Y ) is dualizable.

Mathematical Subject Classification 2000
Primary: 46M05
Published: 1 June 1973
Barry Simon
Department of Mathematics
California Institute of Technology
MC 253-37
Pasadena CA 91125
United States