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Grothendieck's Theorem and operator integral mappings
Author(s) -
Randrianantoanircisse
Publication year - 2013
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jds069
Subject(s) - mathematics , bounded function , pure mathematics , bounded operator , commutative property , hilbert space , bounded inverse theorem , domain (mathematical analysis) , operator (biology) , linear map , discrete mathematics , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene
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: ℓ 1 →ℓ 2 is linear and bounded then α: max(ℓ 1 )→ R + C is exactly integral in the sense of Effros and Ruan. Similarly, we obtain that α: max(ℓ 1 )→min(ℓ 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.