Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T10:19:37.134Z Has data issue: false hasContentIssue false
  • English
  • Français

TENSOR PRODUCTS OF MAXIMAL ABELIAN SUBALGBERAS OF C*-ALGEBRAS

Published online by Cambridge University Press:  01 May 2008

SIMON WASSERMANN
SIMON WASSERMANN*
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that if C1 and C2 are maximal abelian self-adjoint subalgebras (masas) of C*-algebras A1 and A2, respectively, then the completion C1 ⊗ C2 of the algebraic tensor product C1 ⊙ C2 of C1 and C2 in any C*-tensor product A1 ⊗βA2 is maximal abelian provided that C1 has the extension property of Kadison and Singer and C2 contains an approximate identity for A2. Examples are given to show that this result can fail if the conditions on the two masas do not both hold. This gives an answer to a long-standing question, but leaves open some other interesting problems, one of which turns out to have a potentially intriguing implication for the Kadison-Singer extension problem.

Keywords

46L06
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 2 , May 2008 , pp. 209 - 216
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Akemann, C. A. and Shultz, F.Perfect C*-algebras, Mem. Amer. Math. Soc. 55 (1985), no. 326Google Scholar
2.Anderson, J.A maximal abelian subalgebra of the Calkin algebra with the extension property, Math. Scand. 42 (1978), 101–110.CrossRefGoogle Scholar
3.Archbold, R. J.On the centre of a tensor product of C*-algebras, J. London Math. Soc. (2) 10 (1975), 257–262.CrossRefGoogle Scholar
4.Archbold, R. J.Extensions of states of C*-algebras, J. London Math. Soc. (2) 21 (1980), 351–354.CrossRefGoogle Scholar
5.Batty, C. J. K.On relative commutants in tensor products of C*-algebras, Math. Z. 151 (1976), 215–218.CrossRefGoogle Scholar
6.Haydon, R. G. and Wassermann, S.A commutation result for tensor products of C*-algebras, Bull. London Math. Soc. 5 (1973), 283–287.CrossRefGoogle Scholar
7.Junge, M. and Pisier, G.Bilinear forms on exact operator spaces and B(H)⊗ B(H), Geom. Funct. Anal. 5 (1995), 329–363.CrossRefGoogle Scholar
8.Lance, E. C.Tensor products of non-unital C*-algebras, J. London Math. Soc. (2) 12 (1975/76), 160–168.Google Scholar
9.Kadison, R. V. and Singer, I. M.Extensions of pure states, Amer. J. Math. 81 (1959), 383–400.CrossRefGoogle Scholar
10.Sakai, S.C*-algebras and W*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60 (Springer-Verlag, 1971).Google Scholar
11.Takesaki, M.On the cross-norm of the direct product of C*-algebras, Tôhoku Math. J. (2) 16 (1964), 111–122.CrossRefGoogle Scholar
12.Wassermann, S.The slice map problem for C*-algebras, Proc. London Math. Soc. (3) 32 (1976), 537–559.CrossRefGoogle Scholar
13.Wassermann, S.On tensor products of certain group C*-algebras, J. Functional Analysis 23 (1976), 239–254.CrossRefGoogle Scholar
14.Wassermann, S.Tensor products of free-group C*-algebras, Bull. London Math. Soc. 22 (1990), 375–380.CrossRefGoogle Scholar