Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Equivalence of norms on operator space tensor products of $C^*$-algebras


Authors: Ajay Kumar and Allan M. Sinclair
Journal: Trans. Amer. Math. Soc. 350 (1998), 2033-2048
MSC (1991): Primary 46L05; Secondary 46C10, 47035
DOI: https://doi.org/10.1090/S0002-9947-98-02190-4
MathSciNet review: 1473449
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Haagerup norm $\Vert \cdot \Vert _{h}$ on the tensor product $A\otimes B$ of two $C^*$-algebras $A$ and $B$ is shown to be Banach space equivalent to either the Banach space projective norm $\Vert \cdot \Vert _{\gamma }$ or the operator space projective norm $\Vert \cdot \Vert _{\wedge }$ if and only if either $A$ or $B$ is finite dimensional or $A$ and $B$ are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on $A\otimes B$ if and only if $A$ or $B$ is subhomogeneous.


References [Enhancements On Off] (What's this?)

  • 1. R.J. Archbold and C.J.K. Batty, $C^{*}$-tensor norms and slice maps, J. London Math. Soc. (2) 22 (1980) 127-138. MR 81j:46090
  • 2. D.P. Blecher, Geometry of the tensor product of $C^{*}$-algebras, Math. Proc. Camb. Phil. Soc. 104 (1988), 119-127. MR 89g:46094
  • 3. D.P. Blecher, Tensor products of operator spaces II, Can. J. Math. 44 (1992), 75-90. MR 93e:46084
  • 4. D.P. Blecher and V.I. Paulsen, Tensor product of operator spaces, J. Funct. Anal. 99 (1991), 262-292. MR 93d:46095
  • 5. D.P. Blecher and R.R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. 45 (1992), 126-144. MR 93h:46078
  • 6. E. Christensen and A.M. Sinclair, A survey of completely bounded operators, Bull. London Math. Soc. 21 (1989), 417-448. MR 91b:46051
  • 7. E.G. Effros and A. Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36 (1987), 257-276. MR 89b:46068
  • 8. E.G. Effros and Z-J. Ruan, On approximation properties for operator spaces, International J. Math. 1 (1990), 163-187. MR 92g:46089
  • 9. E.G. Effros and Z-J. Ruan, Self duality for the Haagerup tensor product and Hilbert space factorization, J. Funct. Anal. 100 (1991), 257-284. MR 93f:46090
  • 10. E.G. Effros and Z-J. Ruan, A new approach to operator spaces, Bull. Can. Math. Soc. 34 (1991), 329-337. MR 93a:47045
  • 11. E.G. Effros and Z-J. Ruan, Operator convolution algebras. An approach to Quantum groups. (preprint).
  • 12. U. Haagerup, The Grothendieck inequality for bilinear forms on $C^{*}$-algebras, Advances in Math. 56 (1985), 93-116. MR 86j:46061
  • 13. T. Huruya and J. Tomiyama, Completely bounded maps of $C^{*}$-algebras, J. Operator Theory 10 (1983), 141-152. MR 85f:46108a
  • 14. T. Itoh, The Haagerup type cross norm on $C^{*}$-algebras, Proc. Amer. Math. Soc. 109 (1990), 689-695. MR 90m:46096
  • 15. B.E. Johnson, R.V. Kadison and J.R. Ringrose, Cohomology of operator algebras III, Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972),73-96. MR 47:7454
  • 16. S. Kaijser and A.M. Sinclair, Projective tensor products of $ C^{*}$-algebras, Math. Scand. 55 (1984), 161-187. MR 86m:46053
  • 17. A. Kumar, Involution and the Haagerup tensor product (preprint).
  • 18. M. Mathieu, Generalising elementary operators, Semesterbericht Funktionalanalysis SS88, Tübingen (1988), 133-153.
  • 19. V.I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math. 146 Longman, 1986. MR 88h:46111
  • 20. V.I. Paulsen and R.R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987), 258-276. MR 89m:46099
  • 21. G. Pisier, Grothendieck's theorem for non-commutative $C^{*}$-algebras with an appendix on Grothendieck's constant, J. Funct. Anal. 29 (1978), 397-415. MR 80j:47027
  • 22. G. Pisier, Factorization of linear operators and the geometry of Banach spaces, CBMS Series No. 60, A.M.S. Providence, R.I., 1986. MR 88a:47020
  • 23. A.M. Sinclair and R.R. Smith, Hochschild Cohomology of von Neumann Algebras, Lecture Notes Series 203, London Mathematical Society, 1995. MR 96d:46094
  • 24. R.R. Smith, Completely bounded maps between $C^{*}$-algebras, J. London Math. Soc. 27 (1983), 157-166. MR 84g:46086
  • 25. R.R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), 156-175. MR 93a:46115
  • 26. M. Takesaki, Theory of operator algebras I, Springer-Verlag, Berlin, 1979. MR 81e:46038

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46L05, 46C10, 47035

Retrieve articles in all journals with MSC (1991): 46L05, 46C10, 47035


Additional Information

Ajay Kumar
Affiliation: Department of Mathematics, Rajdhani College (University of Delhi), Raja Garden, New Delhi-110015, India

Allan M. Sinclair
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
Email: allan@maths.ed.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-98-02190-4
Keywords: Banach space projective norm, operator space projective norm, Haagerup norm, $C^*$-algebras, second dual
Received by editor(s): August 16, 1996
Additional Notes: Supported by Commonwealth Academic Staff Fellowship at the University of Edinburgh
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society