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Contractive projections and operator spaces

Author(s): Matthew Neal; Bernard Russo
Journal: Trans. Amer. Math. Soc. 355 (2003), 2223-2262.
MSC (2000): Primary 17C65; Secondary 46L07
Posted: January 27, 2003
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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.

References:

1.
J. Arazy and Y. Friedman, Contractive projections in $C_1$ and $C_\infty$, Mem. Amer. Math. Soc. 13 (1978), no 200. MR 82b:47023

2.
T. J. Barton, T. C. Dang and G. Horn, Normal representations of Banach Jordan triple systems, Proc. Amer. Math. Soc. 102 (1988), 551-555. MR 89c:46065

3.
T. J. Barton and R. Timoney, Weak$^*$-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), 177-191. MR 88d:46129

4.
D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1992), 262-292. MR 93d:46095

5.
O. Bratteli and D. W. Robinson, Operator algebras and quantum statistical mechanics II, Springer-Verlag, 1981. MR 82k:82013

6.
M. D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156-209. MR 55:3814

7.
E. Christensen and A. M. Sinclair, Completely bounded isomorphisms of injective von Neumann algebras, Proc. Edinburgh Math. Soc. 32 (1989), 317-327. MR 90k:46135

8.
T. Dang and Y. Friedman, Classification of $JBW^*$-triple factors and applications, Math. Scand. 61 (1987), 292-330. MR 89g:46109

9.
E. Effros and Z. J. Ruan, On matricially normed spaces, Pacific J. Math. 132 (1988), 243-264. MR 90a:46150

10.
E. Effros and Z. J. Ruan, Operator Spaces, Oxford University Press, 2000. MR 2002a:46082

11.
E. Effros and E. Størmer, Positive projections and Jordan structure in operator algebras, Math. Scand. 45 (1979), 127-138. MR 82e:46076

12.
Y. Friedman and B. Russo, Contractive projections on operator triple systems, Math. Scand. 52 (1983), 279-311. MR 84m:46090

13.
Y. Friedman and B. Russo, Solution of the contractive projection problem, J. Funct. Anal. 60 (1985), 56-79. MR 87a:46115

14.
Y. Friedman and B. Russo, Structure of the predual of a $JBW^*$-triple, J. Reine Angew. Math. 356 (1985), 67-89. MR 86f:46073

15.
H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Pitman, 1984. MR 86a:46092

16.
L. A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces. In: T. L. Hayden and T. J. Suffridge (eds.), Infinite dimensional holomorphy. Proceedings, 1973 (Lecture Notes in Mathematics 364, pp. 13-40), Springer-Verlag, 1974. MR 53:11106

17.
M. Hestenes, A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal. 11 (1962), 138-194. MR 27:169

18.
G. Horn, Classification of $JBW^*$-triples of Type I, Math. Zeit. 196 (1987), 271-291. MR 88m:46076

19.
G. Horn, Coordinatization theorems for $JBW^*$-triples, Quart. J. Math. Oxford Ser. (2) 38 (1987), 321-335. MR 88m:46075

20.
G. Horn and E. Neher, Classification of continuous $JBW^*$-triples, Trans. Amer. Math. Soc. 306 (1987), 553-578. MR 89c:46090

21.
W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Zeit. 183 (1983), 503-529. MR 85c:46040

22.
W. Kaup, Contractive projections on Jordan $C^*$-algebras and generalizations, Math. Scand. 54 (1984), 95-100. MR 85h:17012

23.
O. Loos, Jordan Pairs, Lecture Notes in Mathematics 460, Springer-Verlag, 1975. MR 56:3071

24.
O. Loos, Bounded symmetric domains and Jordan pairs, Lecture Notes, Univ. of California, Irvine, 1977.

25.
M. Neal and B. Russo, Contractive projections and operator spaces, C. R. Acad. Sci. Paris 331 (2000), 873-878. MR 2001m:46128

26.
E. Neher, Jordan triple systems by the grid approach, Lecture Notes in Mathematics 1280, Springer, 1987. MR 89b:17024

27.
T. Oikhberg and H. P. Rosenthal, Extension properties for the space of compact operators, J. Funct. Anal. 179 (2001), 251-308. MR 2002b:47138

28.
G. Pisier, The operator Hilbert space ${\rm OH}$, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no 585. MR 97a:46024

29.
G. Pisier, Noncommutative vector-valued $L_p$-spaces and completely $p$-summing maps, Astérisque No. 247, Soc. Math. France, 1998. MR 2000a:46108

30.
A. G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 1, 183-190. MR 93d:46101

31.
A. G. Robertson and S. Wasserman, Completely bounded isomorphisms of injective operator systems, Bull. London Math. Soc. 21 (1989), 285-290. MR 90c:47077
32.
A. G. Robertson and M. A. Youngson, Isomorphisms of injective operator spaces and Jordan triple systems, Quart. J. Math. Oxford Ser. (2) 41 (1990), 449-462. MR 91m:46113

33.
Z.-J. Ruan, Subspaces of $C^*$-algebras, J. Funct. Anal. 76 (1988), 217-230. MR 89h:46082

34.
Z.-J. Ruan, Injectivity of operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89-104. MR 91d:46078

35.
B. Russo, Structure of $JB^*$-triples, In: Jordan Algebras, Proceedings of the Oberwolfach Conference 1992, Eds: W. Kaup, K. McCrimmon, H. Petersson, de Gruyter, Berlin (1994), 209-280. MR 95h:46109

36.
S. Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete 60, Springer-Verlag, 1971. MR 56:1082

37.
H. Upmeier, Symmetric Banach manifolds and Jordan $C^*$-algebras, North-Holland Publishing Co., Amsterdam-New York, 1985. MR 87a:58022

38.
M. A. Youngson, Completely contractive projections on $C^*$-algebras, Quart. J. Math. Oxford Ser. (2) 34 (1983), 507-511. MR 85f:46112

39.
H. Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117-143. MR 84h:46093

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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: 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
Posted: January 27, 2003
Additional Notes: This work was supported in part by NSF grant DMS-0101153
Copyright of article: Copyright 2003, American Mathematical Society

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