Splitting for subalgebras of tensor products

Zacharias, Joachim

doi:10.1090/S0002-9939-00-05629-X

Splitting for subalgebras of tensor products
HTML articles powered by AMS MathViewer

by Joachim Zacharias PDF

Proc. Amer. Math. Soc. 129 (2001), 407-413 Request permission

Abstract:

We prove splitting results for subalgebras of tensor products of operator algebras. In particular, any $C^*$-algebra $C$ s.t. $A\otimes 1 \subseteq C \subseteq A \otimes B$ is a tensor product $A\otimes B_0$ provided $A$ is simple and nuclear.

References

Joel Anderson, Extensions, restrictions, and representations of states on $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 249 (1979), no. 2, 303–329. MR 525675, DOI 10.1090/S0002-9947-1979-0525675-1
Étienne Blanchard, Déformations de $C^*$-algèbres de Hopf, Bull. Soc. Math. France 124 (1996), no. 1, 141–215 (French, with English and French summaries). MR 1395009, DOI 10.24033/bsmf.2278
Etienne Blanchard, Tensor products of $C(X)$-algebras over $C(X)$, Astérisque 232 (1995), 81–92. Recent advances in operator algebras (Orléans, 1992). MR 1372526
Joachim Cuntz, The structure of multiplication and addition in simple $C^*$-algebras, Math. Scand. 40 (1977), no. 2, 215–233. MR 500176, DOI 10.7146/math.scand.a-11691
Jean De Cannière and Uffe Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500. MR 784292, DOI 10.2307/2374423
L. Ge and R. Kadison, On tensor products for von Neumann algebras, Invent. Math. 123 (1996), no. 3, 453–466. MR 1383957, DOI 10.1007/s002220050036
Eberhard Kirchberg, Exact $\textrm {C}^*$-algebras, tensor products, and the classification of purely infinite algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 943–954. MR 1403994
E. Kirchberg: Classification of purely infinite $C^*$-algebras using Kasparov’s theory, preprint (1994)
Eberhard Kirchberg and Simon Wassermann, Operations on continuous bundles of $C^*$-algebras, Math. Ann. 303 (1995), no. 4, 677–697. MR 1359955, DOI 10.1007/BF01461011
M. A. Naimark: Normed rings, Groningen 1972
S. Stratila, L. Zsido: A commutation theorem in tensor products of von Neumann algebras, preprint (1998)
Simon Wassermann, The slice map problem for $C^*$-algebras, Proc. London Math. Soc. (3) 32 (1976), no. 3, 537–559. MR 410402, DOI 10.1112/plms/s3-32.3.537
Simon Wassermann, A pathology in the ideal space of $L(H)\otimes L(H)$, Indiana Univ. Math. J. 27 (1978), no. 6, 1011–1020. MR 511255, DOI 10.1512/iumj.1978.27.27069
L. Zsido: A criterion for splitting $C^*$-algebras in tensor products, to appear in Proc. AMS