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Lifting Problems and the Cohomology of C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Man-Duen Choi  and
Edward G. Effros
Man-Duen Choi
Affiliation:
University of Toronto, Toronto, Ontario
Edward G. Effros
Affiliation:
University of Pennsylvania, Philadelphia, Pennsylvania
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Suppose that A and B are C*-algebras, J is a closed two-sided ideal in B, and that η: B →B/J is the quotient map. Given a linear contraction φ : A →B/J, a linear map Ψ: A →B is a lifting of φ if one has a commutative diagram

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 1092 - 1111
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Alfsen, E. M., Compact convex sets and boundary integrals (Springer Verlag, Berlin, 1971).Google Scholar
2. Alfsen, E. M. and Effros, E. G., Structure in real Banach spaces, Ann. of Math. 96 (1972), 98–173.Google Scholar
3. Andersen, T. B., Linear extensions, projections, and split faces, J. Func. Anal. 17 (1974), 161–173.Google Scholar
4. Arveson, W. B., A note on essentially normal operators, Proc. Royal Irish Acad., Sect. A74 (1974), 143–146.Google Scholar
5. Berger, C. A., L. A. Coburn and Lebow, A., Representation and index theory for C*-algebras generated by commuting isometrics, to appear.Google Scholar
6. Briem, E., Linear extensions and linear liftings in subspaces of C(X), to appear.Google Scholar
7. Brown, L. G., R. G. Douglas and P.A. Fillmore, Unitary equivalence modulo the compact operators and extensions of C*-algebras, Proc. Conf. on Operator Theory, Lecture Notes in Math., Vol. 345 (Springer Verlag, New York, 1973), 58–128.Google Scholar
8. Brown, L. G. Extensions of C*-algebras and K-homology, Ann. of Math. 105 (1977), 265–324.Google Scholar
9. Choi, M. D., A Schwarz inequality for positive linear maps on C*-algebras, 111. J. Math. 18 (1974), 565–574.Google Scholar
11. Choi, M. D. and Effros, E. G., Injectivity and operator spaces, J. Func. Anal. 21 (1977), 156–209.Google Scholar
12. Choi, M. D. and Effros, E. G. Nuclear C*-algebras and the approximation property, Amer. J. Math., to appear.Google Scholar
13. Choi, M. D. and Effros, E. G. Separable nuclear C*-algebras and injectivity, Duke Math. J. 43 (1976), 309–322.Google Scholar
14. Choi, M. D. and Effros, E. G. The completely positive lifting problem for C*-algebras, Ann. of Math. 104 (1976), 585–609.Google Scholar
15. Cohen, P. J., Factorization in group algebras, Duke Math. J. 26 (1959), 199–205.Google Scholar
16. Connes, A., Classification of infective factors, Ann. of Math. 104 (1976), 73–116.Google Scholar
17. Conway, J., The compact operators are not complemented in £$(H), Proc. Amer. Math. Soc. 32 (1972), 549–550.Google Scholar
18. Craw, I. G., Axiomatic cohomology for Banach modules, Proc. Amer. Math. Soc. 38 (1973), 68–74.Google Scholar
19. Craw, I. G. Axiomatic cohomology of operator algebras, Bull. Soc. Math. France 101 (1973), 449–460.Google Scholar
20. Cunningham, F., L-structure in L-spaces, Trans. Amer. Math. Soc. 95 (1960), 274–299.Google Scholar
21. Day, M. M., Normed linear spaces (Springer Verlag, Academic Press, New York, 1962).Google Scholar
22. Dean, D. W., The equation L(E, X**) = L(E, X)** and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146–148.Google Scholar
23. Dixmier, J., Les C*-algèbres et leurs représentations (Gauthier Villars, Paris, 1964).Google Scholar
24. Grothendieck, A., Produits tensoriels topologiques et espaces nucléaire, Mem. Amer. Math. Soc. 16 (Providence, 1955).Google Scholar
25. Grothendieck, A. La théorie de Fredholm, Bull. Soc. Math. France 84 (1956), 319–384.Google Scholar
26. Hennefeld, J., A decomposition for B(X)* and unique Hahn-Banach extensions, Pac. J. Math. 46 (1973), 197–199.Google Scholar
27. Hilton, P. J., Stammbach, U., A course in homologuai algebra, Graduate Texts in Mathematics, No. 4 (Springer Verlag, New York, 1970).Google Scholar
28. Johnson, B. E., Cohomology in Banach algebras, Mem. of Amer. Math. Soc. 127 (Providence, 1972).Google Scholar
29. Johnson, B. E. Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685–698.Google Scholar
30. Johnson, B. E. Perturbations of Banach algebras, to appear.Google Scholar
31. Johnson, B. E., Kadison, R. V., and Ringrose, J. R., Cohomology of operator algebras III: Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972), 73–96.Google Scholar
32. Kadison, R. V., Transformation of states in operator theory and dynamics, Topology 3, Suppl.0 (1965), 177–198.Google Scholar
33. Kadison, R. V. and Ringrose, J. R., Cohomology of operator algebras, I: type I von Neumann algebras. Acta Math. 126 (1971), 227–243.Google Scholar
34. Raeburn, I. and Taylor, J. L., Hochschild cohomology and perturbations of Banach algebras, J. Func. Anal. 25 (1977), 258–266.Google Scholar
35. Ringrose, J. R., Cohomology of operator algebras, Lecture Notes in Math., Vol. 247 (Springer Verlag, New York, 1972).Google Scholar
36. Goullet de Rugy, A., Géométrie des simplexes, Centre de Documentation Universitaire, Paris, 1968.Google Scholar
37. Sinclair, A. M., Annihilator ideals in the cohomology of Banach algebras, Proc. Amer. Math. Soc. 33 (1972), 361–366.Google Scholar
38. Vesterstrøm, J., Positive linear extension operators for spaces of affine functions, Israel J. Math. 16 (1973), 203–211.Google Scholar
39. Wassermann, S., On tensor products of certain group C*-algebras, J. Funct. Anal. 23 (1976), 239–254.Google Scholar