M-complete approximate identities in operator spaces
This work introduces the concept of an M-complete approximate identity (M-cai) for a given operator subspace X of an operator space Y. M-cai's generalize central approximate identities in ideals in C*-algebras, for it is proved that if X admits an M-cai in Y, then X is a complete M-ideal in Y. It is proved, using 'special' M-cai's, that if J is a nuclear ideal in a C*-algebra A, then J is completely complemented in Y for any (isomorphically) locally reflexive operator space Y with J ⊂ Y ⊂ A and Y/J separable. (This generalizes the previously known special case where Y=A , due to Effros-Haagerup.) In turn, this yields a new proof of the Oikhberg-Rosenthal Theorem that K is completely complemented in any separable locally reflexive operator superspace, where K is the C*-algebra of compact operators on $l^2$. M-cai's are also used in obtaining some special affirmative answers to the open problem of whether K is Banach-complemented in A for any separable C*-algebra A with $K ⊂A ⊂ B(l^2)$. It is shown that if, conversely, X is a complete M-ideal in Y, then X admits an M-cai in Y in the following situations: (i) Y has the (Banach) bounded approximation property; (ii) Y is 1-locally reflexive and X is λ-nuclear for some λ ≥ 1; (iii) X is a closed 2-sided ideal in an operator algebra Y (via the Effros-Ruan result that then X has a contractive algebraic approximate identity). However, it is shown that there exists a separable Banach space X which is an M-ideal in Y=X**, yet X admits no M-approximate identity in Y.
- [AP] C. A. Akeman and G. K. Pedersen, Ideal perturbations of elements in C*-algebras, Math. Scand. 41 (1977), 117-139.
- [AE] E. Alfsen and E. Effros, Structure in real Banach spaces, Ann. of Math. 96 (1972), 98-173.
- [A] T. B. Andersen, Linear extensions, projections, and split faces, J. Funct. Anal. 17 (1974), 161-173.
- [An] T. Ando, A theorem on non-empty intersection of convex sets and its applications, J. Approx. Theory 13 (1975), 158-166.
- [Ar] W. Arveson, Notes on extensions of C*-algebras, Duke Math. J. 44 (1977), 329-355.
- [BP] D. P. Blecher and V. I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292.
- [B] K. Borsuk, Über Isomorphie der Funktionalräume, Bull. Int. Acad. Polon. Sci. A 1/3 (1933), 1-10.
- [Be] S. F. Bellenot, Local reflexivity of normed spaces, J. Funct. Anal. 59 (1984), 1-11.
- [CJ] C.-M. Cho and W. B. Johnson, M-ideals and ideals in L(X), J. Operator Theory 16 (1986), 245-260.
- [CE1] M.-D. Choi and E. Effros, The completely positive lifting problem for C*-algebras, Ann. of Math. 104 (1976), 585-609.
- [CE2] M.-D. Choi and E. Effros, Lifting problems and the cohomology of C*-algebras, Canad. J. Math. 29 (1977), 1092-1111.
- [D] K. R. Davidson, C*-Algebras by Example, Amer. Math. Soc., Providence, 1996.
- [DP] K. R. Davidson and S. C. Power, Best approximation in C*-algebras, J. Reine Angew. Math. 368 (1986), 43-62.
- [Di] J. Dixmier, Les fonctionnelles linéaires sur l'ensemble des opérateurs bornés d'un espace de Hilbert, Ann. of Math. 51 (1950), 387-408.
- [Du] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
- [EH] E. Effros and U. Haagerup, Lifting problems and local reflexivity for C*-algebras, Duke Math. J. 52 (1985), 103-128.
- [EOR] E. Effros, N. Ozawa and Z.-J. Ruan, On injectivity and nuclearity for operator spaces, to appear.
- [ER1] E. Effros and Z.-J. Ruan, On non-self-adjoint operator algebras, Proc. Amer. Math. Soc. 110 (1990), 915-922.
- [ER2] E. Effros and Z.-J. Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. 69 (1994), 171-197.
- [GH] L. Ge and D. Hadwin, Ultraproducts for C*-algebras, to appear.
- [GN] I. M. Gelfand and M. A. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sb. (N.S.) 12 (54) (1943), 197-213.
- [HL] P. Harmand and Å. Lima, Banach spaces which are M-ideals in their biduals, Trans. Amer. Math. Soc. 283 (1984), 253-264.
- [HWW] P. Harmand, D. Werner and W. Werner, M-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer, 1993.
- [JO] W. B. Johnson and T. Oikhberg, Separable lifting property and extensions of local reflexivity, to appear.
- [JRZ] W. B. Johnson, H. P. Rosenthal and M. Zippin, On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1977), 488-506.
- [K] N. J. Kalton, M-ideals of compact operators, Illinois J. Math. 37 (1993), 147-169.
- [KW] N. J. Kalton and D. Werner, Property (M), M-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995) 137-178.
- [Ki] E. Kirchberg, On non-semisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math. 112 (1993), 449-489.
- [KR] S.-H. Kye and Z.-J. Ruan, On local lifting property for operator spaces, preprint.
- [L1] Å. Lima, Intersection properties of balls and subspaces of Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62.
- [L2] Å. Lima, The metric approximation property, norm-one projections and intersections of balls, Israel J. Math. 84 (1993), 451-475.
- [LR] J. Lindenstrauss and H. P. Rosenthal, Automorphisms in $c_0$, $l_1$ and m, Israel J. Math. 7 (1969), 222-239.
- [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977.
- [M] E. Michael, Some extension theorems for continuous functions, Pacific J. Math. 3 (1953), 789-806.
- [OR] T. Oikhberg and H. P. Rosenthal, Extension properties for the space of compact operators, J. Funct. Anal., submitted.
- [Oj] E. Oja, On the uniqueness of the norm-preserving extension of a linear functional in the Hahn-Banach Theorem, Izv. Akad. Nauk. Est. SSR 33 (1984), 424-438 (in Russian).
- [O] N. Ozawa, A short proof of the Oikhberg-Rosenthal Theorem, preprint.
- [Pa] V. I. Paulsen, Completely Bounded Maps and Dilations, Pitman Res. Notes Math. Ser. 146, Longman, 1986.
- [PW] R. Payá and W. Werner, An approximation property related to M-ideals of compact operators, Proc. Amer. Math. Soc. 111 (1991), 993-1001.
- [Pe] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 29 (1960), 209-227.
- [Pi] G. Pisier, An introduction to the theory of operator spaces, preprint.
- [R] A. G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991), 183-190.
- [Ro] H. P. Rosenthal, The complete separable extension property, J. Operator Theory 43 (2000), 329-374.
- [S] R. Smith, An addendum to 'M-ideal structure in Banach algebras', J. Funct. Anal. 32 (1979), 269-271.
- [SW] R. Smith and J. Ward, M-ideal structure in Banach algebras, ibid. 27 (1978), 337-349.
- [So] A. Sobczyk, Projection of the space (m) on its subspace $(c_0)$, Bull. Amer. Math. Soc. 47 (1941), 938-947.
- [V] W. A. Veech, Short proof of Sobczyk's Theorem, Proc. Amer. Math. Soc. 28 (1971), 627-628.
- [W1] D. Werner, M-structure in tensor products of Banach spaces, Math. Scand. 61 (1987), 149-164.
- [W2] D. Werner, Remarks on M-ideals of compact operators, Quart. J. Math. Oxford (2) 41 (1990), 501-507.
- [We] W. Werner, Inner M-ideals in Banach algebras, Math. Ann. 291 (1991), 205-223.
- [Z] M. Zippin, The separable extension problem, Israel J. Math. 26 (1977), 372-387