Grothendieck's Theorem and operator integral mappings

摘要

We prove that bounded linear mappings from L_1-spaces into Hilbert spaces satisfy integral-type properties when viewed as completely bounded mappings. More precisely, we show that if α: ?_1→?_2 is linear and bounded then α: max(?1)→R+C is exactly integral in the sense of Effros and Ruan. Similarly, we obtain that α: max(?1) →min(?_2) is completely integral. We also discuss integrability properties when the domain is a non-commutative L 1-space. These results may be viewed as variants of Grothendieck's Theorem in the category of operator spaces.