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Operator Algebras with Unique Preduals

Published online by Cambridge University Press:  20 November 2018

Kenneth R. Davidson  and
Alex Wright
Kenneth R. Davidson
Affiliation:
Pure Math. Dept., U. Waterloo, Waterloo, ON N2L–3G1 e-mail: [email protected]@gmail.com
Alex Wright
Affiliation:
Pure Math. Dept., U. Waterloo, Waterloo, ON N2L–3G1 e-mail: [email protected]@gmail.com
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Abstract

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We show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-$*$ closed unital operator algebra containing a weak-$*$ dense subalgebra of compact operators has a unique Banach space predual.

Keywords

47L5046B0447L35unique predualfree semigroup algebraCSL algebra
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 411 - 421
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Ando, T., On the predual of H1 .. Comment. Math. Special Issue 1(1978), 33–40.Google Scholar
[2] Davidson, K. R., Nest algebras. Pitman Research Notes in Mathematics Series, 191, Longman Scientific and Technical, Harlow; JohnWiley & Sons, New York, 1988.Google Scholar
[3] Davidson, K. R., Free semigroup algebras, a survey. In: Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl., 129, Birkhauser, Basel, 2001, pp. 209–240.Google Scholar
[4] Davidson, K. R., Katsoulis, E., and Pitts, D. R., The structure of free semigroup algebras.. J. Reine Angew. Math. 533(2001), 99–125.Google Scholar
[5] Davidson, K. R. and Pitts, D. R., Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. 78(1999), no. 2, 401–430. doi:10.1112/S002461159900180XGoogle Scholar
[6] Davidson, K. R. and Pitts, D. R., The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311(1998), no. 2, 275–303. doi:10.1007/s002080050188Google Scholar
[7] Davidson, K. R. and Power, S. C., Best approximation in C*-algebras.. J. Reine Angew. Math. 368(1986), 43–62.Google Scholar
[8] Effros, E. G., Ozawa, N., and Ruan, Z.-J., On injectivity and nuclearity for operator spaces. Duke Math. J. 110(2001), no. 3, 489–521. doi:10.1215/S0012-7094-01-11032-6Google Scholar
[9] Erdos, J. A., Operators of finite rank in nest algebras.. J. London Math. Soc. 43(1968), 391–397. doi:10.1112/jlms/s1-43.1.391Google Scholar
[10] Godefroy, G., Existence and uniqueness of isometric preduals: a survey. In: Banach space theory (Iowa City, IA, 1987), Contemp. Math., 85, American Mathematical Society, Providence, RI, 1989.Google Scholar
[11] Godefroy, G. and Li, D., Banach spaces which are M-ideals in their bidual have property (u) . Ann. Inst. Fourier (Grenoble) 39(1989), no. 2, 361–371.Google Scholar
[12] Godefroy, G. and Saphar, P. D., Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math. 32(1988), no. 4, 672–695.Google Scholar
[13] Godefroy, G. and Talagrand, M., Nouvelles classes d’espaces de Banach à predual unique, Séminaire d’analyse fonctionelle de l’école Polytechnique (1980–81).Google Scholar
[14] Grothendieck, A., Une caractérisation vectorielle-métrique des espaces L1 .. Canad. J. Math. 7(1955), 552–561.Google Scholar
[15] Harmand, P., Werner, D., and Werner, W., M-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics, 1547, Springer-Verlag, Berlin, 1993.Google Scholar
[16] Laurie, C. and Longstaff, W., A note on rank-one operators in reflexive algebras. Proc. Amer. Math. Soc. 89(1983), no. 2, 293–297.Google Scholar
[17] Pfitzner, H., Separable L-embedded Banach spaces are unique preduals. Bull. Lond. Math. Soc. 39(2007), no. 6, 1039–1044. doi:10.1112/blms/bdm077Google Scholar
[18] Popescu, G., Multi-analytic operators and some factorization theorems. Indiana Univ. Math. J. 38(1989), no. 3, 693–710. doi:10.1512/iumj.1989.38.38033Google Scholar
[19] Popescu, G., Multi-analytic operators on Fock spaces. Math. Ann. 303(1995), no. 1, 31–46. doi:10.1007/BF01460977Google Scholar
[20] Ruan, Z.-J., On the predual of dual algebras. J. Operator Theory 27(1992), no. 2, 179–192.Google Scholar
[21] Sakai, S., A characterization of W*-algebras.. Pacific J. Math. 6(1956), 763–773.Google Scholar