Tensor products over abelian 𝑊*-algebras

Magajna, Bojan

doi:10.1090/S0002-9947-96-01655-8

Tensor products over abelian $W^*$-algebras
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Trans. Amer. Math. Soc. 348 (1996), 2427-2440 Request permission

Abstract:

Tensor products of C$^*-$algebras over an abelian W$^*-$algebra $Z$ are studied. The minimal C$^*-$norm on $A\odot _ZB$ is shown to be just the quotient of the minimal C$^*-$norm on $A\odot B$ if $A$ or $B$ is exact.

References

Pere Ara and Martin Mathieu, On the central Haagerup tensor product, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 161–174. MR 1258040, DOI 10.1017/S0013091500018782
Michel Baillet, Yves Denizeau, and Jean-François Havet, Indice d’une espérance conditionnelle, Compositio Math. 66 (1988), no. 2, 199–236 (French). MR 945550
E. Blanchard, Deformations de C$^*-$algebres de Hopf, to appear in Bull. Soc. Math. France.
E. Blanchard, Tensor products of $C(X)-$algebras over $C(X)$, to appear in Asterisque.
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928, DOI 10.2140/pjm.1977.71.349
Avi Chatterjee and Roger R. Smith, The central Haagerup tensor product and maps between von Neumann algebras, J. Funct. Anal. 112 (1993), no. 1, 97–120. MR 1207938, DOI 10.1006/jfan.1993.1027
Edward G. Effros and E. Christopher Lance, Tensor products of operator algebras, Adv. Math. 25 (1977), no. 1, 1–34. MR 448092, DOI 10.1016/0001-8708(77)90085-8
George A. Elliott, Finite projections in tensor product von Neumann algebras, Trans. Amer. Math. Soc. 212 (1975), 47–60. MR 377535, DOI 10.1090/S0002-9947-1975-0377535-3
Thierry Giordano and James A. Mingo, Approximately inner automorphisms of semi-finite von Neumann algebras, Math. Scand. 72 (1993), no. 1, 131–147. MR 1226001, DOI 10.7146/math.scand.a-12441
T. Giordano and J. Mingo, Tensor products of C$^*-$algebras over abelian subalgebras, to appear in J. London Math. Soc.
James Glimm, A Stone-Weierstrass theorem for $C^{\ast }$-algebras, Ann. of Math. (2) 72 (1960), 216–244. MR 116210, DOI 10.2307/1970133
Herbert Halpern, Irreducible module homomorphisms of a von Neumann algebra into its center, Trans. Amer. Math. Soc. 140 (1969), 195–221. MR 241987, DOI 10.1090/S0002-9947-1969-0241987-7
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI 10.1007/BF01404917
E. Kirchberg and S. Wassermann, Operations on continuous bundles of C$^*-$algebras, Math. Ann. 303 (1995), 677–697.
E. C. Lance, Hilbert C$^*-$modules, London Math. Soc. Lecture Note Series 210, Cambridge Univ. Press, Cambridge, 1995.
B. Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal. 129 (1995), 325–348.
B. Magajna, Hilbert modules and tensor products of operator spaces, to appear in Banach Center Publications, Warsaw.
Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
William L. Paschke, Inner product modules over $B^{\ast }$-algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. MR 355613, DOI 10.1090/S0002-9947-1973-0355613-0
Marc A. Rieffel, Morita equivalence for $C^{\ast }$-algebras and $W^{\ast }$-algebras, J. Pure Appl. Algebra 5 (1974), 51–96. MR 367670, DOI 10.1016/0022-4049(74)90003-6
Louis H. Rowen, Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Academic Press, Inc., Boston, MA, 1988. MR 940245
Jean-Luc Sauvageot, Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory 9 (1983), no. 2, 237–252 (French). MR 703809
Jean-Luc Sauvageot, Produits tensoriels de ${\scr Z}$-modules et applications, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 468–485 (French). MR 799587, DOI 10.1007/BFb0074903
Masamichi Takesaki, On the cross-norm of the direct product of $C^{\ast }$-algebras, Tohoku Math. J. (2) 16 (1964), 111–122. MR 165384, DOI 10.2748/tmj/1178243737
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728, DOI 10.1007/978-1-4612-6188-9
N. E. Wegge-Olsen, $K$-theory and $C^*$-algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. A friendly approach. MR 1222415
Takehiko Yamanouchi, Duality for actions and coactions of measured groupoids on von Neumann algebras, Mem. Amer. Math. Soc. 101 (1993), no. 484, vi+109. MR 1127115, DOI 10.1090/memo/0484