The splitting problem for subspaces of tensor products of operator algebras
Author:
Jon Kraus
Journal:
Proc. Amer. Math. Soc. 132 (2004), 11251131
MSC (2000):
Primary 46L10
Published electronically:
November 4, 2003
MathSciNet review:
2045429
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Abstract: The main result of this paper is that if is a von Neumann algebra that is a factor and has the weak* operator approximation property (the weak* OAP), and if is a von Neumann algebra, then every weakly closed subspace of that is an bimodule (under multiplication) splits, in the sense that there is a weakly closed subspace of such that . Note that if is a von Neumann subalgebra of , then is an bimodule if and only if . So this result is a generalization (in the case where has the weak* OAP) of the result of Ge and Kadison that if is a factor, then every von Neumann subalgebra of that contains splits. We also obtain other results concerning the splitting of weakly closed subspaces of tensor products of von Neumann algebras and the splitting of normed closed subspaces of C*algebras that generalize results previously obtained for von Neumann subalgebras and C*subalgebras.
 [1]
Edward
G. Effros, Narutaka
Ozawa, and ZhongJin
Ruan, On injectivity and nuclearity for operator spaces, Duke
Math. J. 110 (2001), no. 3, 489–521. MR 1869114
(2002k:46151), http://dx.doi.org/10.1215/S0012709401110326
 [2]
Edward
G. Effros and ZhongJin
Ruan, On approximation properties for operator spaces,
Internat. J. Math. 1 (1990), no. 2, 163–187. MR 1060634
(92g:46089), http://dx.doi.org/10.1142/S0129167X90000113
 [3]
Edward
G. Effros and ZhongJin
Ruan, Operator spaces, London Mathematical Society Monographs.
New Series, vol. 23, The Clarendon Press, Oxford University Press, New
York, 2000. MR
1793753 (2002a:46082)
 [4]
L.
Ge and R.
Kadison, On tensor products for von Neumann algebras, Invent.
Math. 123 (1996), no. 3, 453–466. MR 1383957
(97c:46074), http://dx.doi.org/10.1007/s002220050036
 [5]
Herbert
Halpern, Module homomorphisms of a von Neumann
algebra into its center, Trans. Amer. Math.
Soc. 140 (1969),
183–193. MR 0241986
(39 #3321), http://dx.doi.org/10.1090/S00029947196902419865
 [6]
Jon
Kraus, The slice map problem for
𝜎weakly closed subspaces of von Neumann algebras, Trans. Amer. Math. Soc. 279 (1983), no. 1, 357–376. MR 704620
(85e:46036), http://dx.doi.org/10.1090/S00029947198307046200
 [7]
Jon
Kraus, The slice map problem and approximation properties, J.
Funct. Anal. 102 (1991), no. 1, 116–155. MR 1138840
(92m:47083), http://dx.doi.org/10.1016/00221236(91)90138U
 [8]
Şerban
Strătilă, Modular theory in operator algebras,
Editura Academiei Republicii Socialiste România, Bucharest; Abacus
Press, Tunbridge Wells, 1981. Translated from the Romanian by the author.
MR 696172
(85g:46072)
 [9]
Şerban
Strătilă and László
Zsidó, An algebraic reduction theory for 𝑊*
algebras. I, J. Functional Analysis 11 (1972),
295–313. MR 0341125
(49 #5875)
 [10]
Şerban
Strătilă and László
Zsidó, The commutation theorem for tensor products over von
Neumann algebras, J. Funct. Anal. 165 (1999),
no. 2, 293–346. MR 1698940
(2000j:46115), http://dx.doi.org/10.1006/jfan.1999.3408
 [11]
Simon
Wassermann, The slice map problem for 𝐶*algebras,
Proc. London Math. Soc. (3) 32 (1976), no. 3,
537–559. MR 0410402
(53 #14152)
 [12]
Simon
Wassermann, On tensor products of certain group
𝐶*algebras, J. Functional Analysis 23
(1976), no. 3, 239–254. MR 0425628
(54 #13582)
 [13]
Simon
Wassermann, Injective 𝑊*algebras, Math. Proc.
Cambridge Philos. Soc. 82 (1977), no. 1, 39–47.
MR
0448108 (56 #6418)
 [14]
Simon
Wassermann, A pathology in the ideal space of
𝐿(𝐻)⊗𝐿(𝐻), Indiana Univ. Math.
J. 27 (1978), no. 6, 1011–1020. MR 511255
(80d:46113), http://dx.doi.org/10.1512/iumj.1978.27.27069
 [15]
Joachim
Zacharias, Splitting for subalgebras of tensor
products, Proc. Amer. Math. Soc.
129 (2001), no. 2,
407–413. MR 1706957
(2001e:46106), http://dx.doi.org/10.1090/S000299390005629X
 [16]
László
Zsidó, A criterion for splitting
𝐶*algebras in tensor products, Proc.
Amer. Math. Soc. 128 (2000), no. 7, 2001–2006. MR 1654056
(2000m:46119), http://dx.doi.org/10.1090/S0002993999052697
 [1]
 E. G. Effros, N. Ozawa, and Z.J. Ruan, On injectivity and nuclearity for operator spaces, Duke Math. J. 110 (2001), 489521. MR 2002k:46151
 [2]
 E. G. Effros and Z.J. Ruan, On approximation properties for operator spaces, International J. Math. 1 (1990), 163187. MR 92g:46089
 [3]
 E. Effros and Z.J. Ruan, Operator Spaces, The Clarendon Press, Oxford University Press, New York, 2000. MR 2002a:46082
 [4]
 L. Ge and R. V. Kadison, On tensor products for von Neumann algebras, Invent. Math. 123 (1996), 453466. MR 97c:46074
 [5]
 H. Halpern, Module homomorphisms of a von Neumann algebra into its center, Trans. Amer. Math. Soc. 140 (1969), 183193. MR 39:3321
 [6]
 J. Kraus, The slice map problem for weakly closed subspaces of von Neumann algebras, Trans. Amer. Math. Soc. 279 (1983), 357376. MR 85e:46036
 [7]
 J. Kraus, The slice map problem and approximation properties, J. Funct. Anal. 102 (1991), 116155. MR 92m:47083
 [8]
 S. Stratila, Modular theory in operator algebras, Abacus Press, Tunbridge Wells, 1981. MR 85g:46072
 [9]
 S. Stratila and L. Zsidó, An algebraic reduction theory for W*algebras, I, J. Funct. Anal. 11 (1972), 295313. MR 49:5875
 [10]
 S. Stratila and L. Zsidó, The commutation theorem for tensor products over von Neumann algebras, J. Funct. Anal. 165 (1999), 293346. MR 2000j:46115
 [11]
 S. Wassermann, The slice map problem for C*algebras, Proc. London Math. Soc. (3) 32 (1976), 537559. MR 53:14152
 [12]
 S. Wassermann, On tensor products of certain group C*algebras, J. Funct. Anal. 23 (1976), 239254. MR 54:13582
 [13]
 S. Wassermann, Injective W*algebras, Math. Proc. Cambridge Philos. Soc. 82 (1977), 3947. MR 56:6418
 [14]
 S. Wassermann, A pathology in the ideal space of , Indiana Univ. Math. J. 27 (1978), 10111020. MR 80d:46113
 [15]
 J. Zacharias, Splitting for subalgebras of tensor products, Proc. Amer. Math. Soc. 129 (2001), 407413. MR 2001e:46106
 [16]
 L. Zsidó, A criterion for splitting C*algebras in tensor products, Proc. Amer. Math. Soc. 128 (2000), 20012006. MR 2000m:46119
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Additional Information
Jon Kraus
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 142602900
Email:
mthjek@acsu.buffalo.edu
DOI:
http://dx.doi.org/10.1090/S0002993903072435
PII:
S 00029939(03)072435
Received by editor(s):
June 14, 2002
Received by editor(s) in revised form:
December 13, 2002
Published electronically:
November 4, 2003
Communicated by:
David R. Larson
Article copyright:
© Copyright 2003
American Mathematical Society
