## Injectivity of operator spaces

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- by Zhong-Jin Ruan PDF
- Trans. Amer. Math. Soc.
**315**(1989), 89-104 Request permission

## Abstract:

We study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces. We give an easy example of an injective operator space which is not completely isometric to any ${C^\ast }$-algebra. This answers a question of Wittstock [23]. Furthermore, we show that an operator space $E$ is injective if and only if there exists an injective ${C^\ast }$-algebra $A$ and two projections $p$ and $q$ in $A$ such that $E$ is completely isometric to $pAq$.## References

- William B. Arveson,
*Subalgebras of $C^{\ast }$-algebras*, Acta Math.**123**(1969), 141–224. MR**253059**, DOI 10.1007/BF02392388 - W. G. Bade,
*The Banach space $C(S)$*, Lecture Notes Series, No. 26, Aarhus Universitet, Matematisk Institut, Aarhus, 1971. MR**0287293** - Man Duen Choi and Edward G. Effros,
*Injectivity and operator spaces*, J. Functional Analysis**24**(1977), no. 2, 156–209. MR**0430809**, DOI 10.1016/0022-1236(77)90052-0 - Henry B. Cohen,
*Injective envelopes of Banach spaces*, Bull. Amer. Math. Soc.**70**(1964), 723–726. MR**184060**, DOI 10.1090/S0002-9904-1964-11189-7 - Edward G. Effros,
*On multilinear completely bounded module maps*, Operator algebras and mathematical physics (Iowa City, Iowa, 1985) Contemp. Math., vol. 62, Amer. Math. Soc., Providence, RI, 1987, pp. 479–501. MR**878396**, DOI 10.1090/conm/062/878396 - Edward G. Effros and Zhong-Jin Ruan,
*On matricially normed spaces*, Pacific J. Math.**132**(1988), no. 2, 243–264. MR**934168** - Dwight B. Goodner,
*Projections in normed linear spaces*, Trans. Amer. Math. Soc.**69**(1950), 89–108. MR**37465**, DOI 10.1090/S0002-9947-1950-0037465-6
U. Haagerup, - Masamichi Hamana,
*Injective envelopes of $C^{\ast }$-algebras*, J. Math. Soc. Japan**31**(1979), no. 1, 181–197. MR**519044**, DOI 10.2969/jmsj/03110181 - Morisuke Hasumi,
*The extension property of complex Banach spaces*, Tohoku Math. J. (2)**10**(1958), 135–142. MR**100781**, DOI 10.2748/tmj/1178244708 - John R. Isbell,
*Three remarks on injective envelopes of Banach spaces*, J. Math. Anal. Appl.**27**(1969), 516–518. MR**251512**, DOI 10.1016/0022-247X(69)90131-0 - J. L. Kelley,
*Banach spaces with the extension property*, Trans. Amer. Math. Soc.**72**(1952), 323–326. MR**45940**, DOI 10.1090/S0002-9947-1952-0045940-5 - Richard I. Loebl,
*Contractive linear maps on $C^*$-algebras*, Michigan Math. J.**22**(1975), no. 4, 361–366 (1976). MR**397423** - Leopoldo Nachbin,
*A theorem of the Hahn-Banach type for linear transformations*, Trans. Amer. Math. Soc.**68**(1950), 28–46. MR**32932**, DOI 10.1090/S0002-9947-1950-0032932-3 - Vern I. Paulsen,
*Completely bounded maps on $C^{\ast }$-algebras and invariant operator ranges*, Proc. Amer. Math. Soc.**86**(1982), no. 1, 91–96. MR**663874**, DOI 10.1090/S0002-9939-1982-0663874-4 - Vern I. Paulsen,
*Every completely polynomially bounded operator is similar to a contraction*, J. Funct. Anal.**55**(1984), no. 1, 1–17. MR**733029**, DOI 10.1016/0022-1236(84)90014-4 - Vern I. Paulsen,
*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472** - Zhong-Jin Ruan,
*Subspaces of $C^*$-algebras*, J. Funct. Anal.**76**(1988), no. 1, 217–230. MR**923053**, DOI 10.1016/0022-1236(88)90057-2 - R. R. Smith,
*Completely bounded maps between $C^{\ast }$-algebras*, J. London Math. Soc. (2)**27**(1983), no. 1, 157–166. MR**686514**, DOI 10.1112/jlms/s2-27.1.157 - W. Forrest Stinespring,
*Positive functions on $C^*$-algebras*, Proc. Amer. Math. Soc.**6**(1955), 211–216. MR**69403**, DOI 10.1090/S0002-9939-1955-0069403-4 - Jun Tomiyama,
*On the transpose map of matrix algebras*, Proc. Amer. Math. Soc.**88**(1983), no. 4, 635–638. MR**702290**, DOI 10.1090/S0002-9939-1983-0702290-4 - Gerd Wittstock,
*Ein operatorwertiger Hahn-Banach Satz*, J. Functional Analysis**40**(1981), no. 2, 127–150 (German, with English summary). MR**609438**, DOI 10.1016/0022-1236(81)90064-1
—, - M. A. Youngson,
*Completely contractive projections on $C^{\ast }$-algebras*, Quart. J. Math. Oxford Ser. (2)**34**(1983), no. 136, 507–511. MR**723287**, DOI 10.1093/qmath/34.4.507

*Decomposition of completely bounded maps on operator algebras*(unpublished), 1980.

*Extensions of completely bounded*${C^\ast }$-

*module homomorphisms*, Proc. Conference on Operator Algebras and Group Representations (Neptun, 1980), Pitman, 1984.

## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**315**(1989), 89-104 - MSC: Primary 46L05; Secondary 47D15
- DOI: https://doi.org/10.1090/S0002-9947-1989-0929239-3
- MathSciNet review: 929239