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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Contractive projections and operator spaces


Authors: Matthew Neal and Bernard Russo
Journal: Trans. Amer. Math. Soc. 355 (2003), 2223-2262
MSC (2000): Primary 17C65; Secondary 46L07
Published electronically: January 27, 2003
MathSciNet review: 1973989
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Abstract: Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces $H_n^k$, $1\le k\le n$, generalizing the row and column Hilbert spaces $R_n$ and $C_n$, and we show that an atomic subspace $X\subset B(H)$ that is the range of a contractive projection on $B(H)$is isometrically completely contractive to an $\ell^\infty$-sum of the $H_n^k$ and Cartan factors of types 1 to 4. In particular, for finite-dimensional $X$, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w$^*$-closed $JW^*$-triples without an infinite-dimensional rank 1 w$^*$-closed ideal.


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Additional Information

Matthew Neal
Affiliation: Department of Mathematics, Denison University, Granville, Ohio 43023
Email: nealm@denison.edu

Bernard Russo
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: brusso@math.uci.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03233-1
PII: S 0002-9947(03)03233-1
Keywords: Contractive projection, operator space, complete contraction, Cartan factor, injective, mixed-injective, $JC^*$-triple, $JW^*$-triple, ternary algebra
Received by editor(s): June 20, 2002
Published electronically: January 27, 2003
Additional Notes: This work was supported in part by NSF grant DMS-0101153
Article copyright: © Copyright 2003 American Mathematical Society