# zbMATH — the first resource for mathematics

Lifting problems and local reflexivity for $$C^ *$$-algebras. (English) Zbl 0613.46047
In [Ann. Math., II. Ser. 104, 585-609 (1976; Zbl 0361.46067)], M.-D. Choi and E. G. Effros proved that if A is an operator system (a unital linear self-adjoint subspace of operators on a Hilbert space), B is a unital $$C^*$$-algebra and J is a closed two-sided ideal in B, then one obstruction to the existence of a completely positive lifting of a completely positive map $$\phi$$ :A$$\to B/J$$ is the irregular behavior of minimal tensor products. Specifically, the obstruction is caused by the fact that the kernel of the map $$A\otimes_{\min}B\to A\otimes_{\min}(B/J)$$ may not equal $$A\otimes_{\min}J$$. In the paper under review, the authors prove that the above obstruction is the only one, by proving the following theorem:
Suppose that $$J$$ is a nuclear ideal in a unital $$C^*$$-algebra $$B$$. Then the following are equivalent:
(1) One can always lift a completely positive unital map $$\phi$$ :A$$\to B/J$$ to a completely positive unital map $$\psi:A\to B$$ for any separable unital $$C^*$$-algebra $$A$$.
(2) The kernel of $$B\otimes_{\min}C\to (B/J)\otimes_{\min}C$$ is $$J\otimes_{\min}C$$ for all unital $$C^*$$-algebras $$C$$.
A $$C^*$$-algebra $$A$$ is said to be approximately injective if given finite-dimensional operator systems $$E_ 1\subseteq E_ 2\subseteq {\mathcal B}({\mathcal H})$$, any completely positive map $$\phi_ 1:E_ 1\to A$$ has completely positive approximate extensions $$\phi_ 2:E_ 2\to A$$, meaning that given $$\epsilon >0$$, there exists a completely positive $$\phi_ 2:E_ 2\to A$$ such that $$\| \phi_ 2|_{E_ 1}- \phi_ 1\| <\epsilon$$. The theorem above is actually proven under the weaker hypothesis that J is approximately injective, although it is unknown whether any hypothesis on J is necessay.
The authors then consider the property of approximate injectivity, prove that it is weaker than nuclearity or (trivially) injectivity, and that it is preserved under direct limits.
Finally, the authors consider a condition on a $$C^*$$-algebra B that implies condition (2) of the main theorem, for all ideals J of B. The condition is a variant of one considered by R. J. Archbold and C. J. K. Batty in [J. London Math., II. Ser. 22, 127-138 (1980; Zbl 0437.46049)], and is also a $$C^*$$-analog of local reflexivity. Let B be a unital $$C^*$$-algebra. Then (local reflexivity) for any finite- dimensional operator system E, any completely positive unital map $${\bar \phi}: E\to B^{**}$$ can be approximated by completely positive unital maps $$\phi: E\to B$$ in the point-weak* topology if and only if, for any $$C^*$$-algebra C, the algebraic tensor product $$B^{**}\otimes C$$, under a natural imbedding into $$(B\otimes_{\min}C)^{**}$$, inherits the minimal norm (Theorem 5.1). These conditions then imply (Proposition 5.3) that for any ideal J in B, kernel (B$$\otimes_{\min}C\to (B/J)\otimes_{\min}C)=J\otimes_{\min}C$$, and in fact a partial converse is proven also (Proposition 5.5).
Reviewer: E.Gootman

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
Full Text:
##### References:
  J. Anderson, A $$C^*$$-algebra $$\mathcal A$$ for which $$\mathrm Ext(\mathcal A)$$ is not a group , Ann. of Math. (2) 107 (1978), no. 3, 455-458. JSTOR: · Zbl 0378.46057 · doi:10.2307/1971124 · links.jstor.org  R. J. Archbold and C. J. K. Batty, $$C^*$$-tensor norms and slice maps , J. London Math. Soc. (2) 22 (1980), no. 1, 127-138. · Zbl 0437.46049 · doi:10.1112/jlms/s2-22.1.127  W. Arveson, Subalgebras of $$C^\ast$$-algebras , Acta Math. 123 (1969), 141-224. · Zbl 0194.15701 · doi:10.1007/BF02392388  W. Arveson, A note on essentially normal operators , Proc. Roy. Irish Acad. Sect. A 74 (1974), 143-146. · Zbl 0311.47010  W. Arveson, Notes on extensions of $$C^*$$-algebras , Duke Math. J. 44 (1977), no. 2, 329-355. · Zbl 0368.46052 · doi:10.1215/S0012-7094-77-04414-3  C. A. Berger, L. A. Coburn, and A. Lebow, Representation and index theory for $$C^*$$-algebras generated by commuting isometries , J. Functional Analysis 27 (1978), no. 1, 51-99. · Zbl 0383.46010 · doi:10.1016/0022-1236(78)90019-8  J. de Canniere and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups , preprint, Odense University 1982, to appear in Amer. J. Math. JSTOR: · Zbl 0577.43002 · doi:10.2307/2374423 · links.jstor.org  M. D. Choi, A simple $$C^\ast$$-algebra generated by two finite-order unitaries , Canad. J. Math. 31 (1979), no. 4, 867-880. · Zbl 0441.46047 · doi:10.4153/CJM-1979-082-4  M. D. Choi and E. Effros, Injectivity and operator spaces , J. Functional Analysis 24 (1977), no. 2, 156-209. · Zbl 0341.46049 · doi:10.1016/0022-1236(77)90052-0  M. D. Choi and E. Effros, Nuclear $$C^*$$-algebras and the approximation property , Amer. J. Math. 100 (1978), no. 1, 61-79. JSTOR: · Zbl 0397.46054 · doi:10.2307/2373876 · links.jstor.org  M. D. Choi and E. Effros, Nuclear $$C^*$$-algebras and injectivity: the general case , Indiana Univ. Math. J. 26 (1977), no. 3, 443-446. · Zbl 0378.46052 · doi:10.1512/iumj.1977.26.26034  M. D. Choi and E. Effros, The completely positive lifting problem for $$C^*$$-algebras , Ann. of Math. (2) 104 (1976), no. 3, 585-609. JSTOR: · Zbl 0361.46067 · doi:10.2307/1970968 · links.jstor.org  M. D. Choi and E. Effros, Lifting problems and the cohomology of $$C^*$$-algebras , Canad. J. Math. 29 (1977), no. 5, 1092-1111. · Zbl 0374.46053 · doi:10.4153/CJM-1977-108-x  D. W. Dean, The equation $$L(E,\,X^\ast\ast)=L(E,\,X)^\ast\ast$$ and the principle of local reflexivity , Proc. Amer. Math. Soc. 40 (1973), 146-148. · Zbl 0263.46014 · doi:10.2307/2038653  J. Dixmier, Les $$C^*$$-algebres et leurs representations , Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Paris, 1996.  R. G. Douglas and R. Howe, On the $$C^*$$-algebra of Toeplitz operators on the quarterplane , Trans. Amer. Math. Soc. 158 (1971), 203-217. · Zbl 0224.47015 · doi:10.2307/1995782  N. Dunford and J. Schwartz, Linear Operators. I. General Theory , With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York, 1958. · Zbl 0084.10402  E. Effros, Aspects of noncommutative order , $$\mathrm C^\ast$$-algebras and applications to physics (Proc. Second Japan-USA Sem., Los Angeles, Calif., 1977), Lecture Notes in Math., vol. 650, Springer, Berlin, 1978, pp. 1-40. · Zbl 0397.46055  E. Effros and L. C. Lance, Tensor products of operator algebras , Adv. Math. 25 (1977), no. 1, 1-34. · Zbl 0372.46064 · doi:10.1016/0001-8708(77)90085-8  A. Guichardet, Tensor products of $$C^\ast$$-algebras Part I , Lecture Notes, vol. 12, Aarhus University, Aarhus, 1969. · Zbl 0228.46056  C. Lance, On nuclear $$C^*$$-algebras , J. Functional Analysis 12 (1973), 157-176. · Zbl 0252.46065 · doi:10.1016/0022-1236(73)90021-9  J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I , Springer-Verlag, Berlin, 1977. · Zbl 0362.46013  V. I. Paulsen, Completely bounded maps on $$C^*$$-algebras and invariant operator ranges , Proc. Amer. Math. Soc. 86 (1982), no. 1, 91-96. · Zbl 0554.46028 · doi:10.2307/2044404  S. Wassermann, On tensor products of certain group $$C^*$$-algebras , J. Functional Analysis 23 (1976), no. 3, 239-254. · Zbl 0358.46040 · doi:10.1016/0022-1236(76)90050-1  S. Wassermann, Liftings in $$C^*$$-algebras: a counterexample , Bull. London Math. Soc. 9 (1977), no. 2, 201-202. · Zbl 0373.46072 · doi:10.1112/blms/9.2.201  G. Wittstock, Ein operatorwertiger Hahn-Banach Satz , J. Funct. Anal. 40 (1981), no. 2, 127-150. · Zbl 0495.46005 · doi:10.1016/0022-1236(81)90064-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.