On biduals of $C^ *$-tensor products
HTML articles powered by AMS MathViewer
- by John C. Quigg PDF
- Proc. Amer. Math. Soc. 100 (1987), 666-668 Request permission
Abstract:
Huruya [4] has proven that, for ${C^*}$-algebras ${A_1}$ and ${A_2}$, \[ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}\] for every ${A_2}$ if and only if ${A_1}$ is scattered. We strengthen this by proving that ${({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}$ if and only if ${A_1}$ or ${A_2}$ is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of ${W^*}$-tensor products.References
- R. J. Archbold, On the centre of a tensor product of $C^\ast$-algebras, J. London Math. Soc. (2) 10 (1975), 257–262. MR 402512, DOI 10.1112/jlms/s2-10.3.257
- R. J. Archbold and C. J. K. Batty, $C^{\ast }$-tensor norms and slice maps, J. London Math. Soc. (2) 22 (1980), no. 1, 127–138. MR 579816, DOI 10.1112/jlms/s2-22.1.127
- Claudio D’Antoni and Roberto Longo, Interpolation by type $\textrm {I}$ factors and the flip automorphism, J. Functional Analysis 51 (1983), no. 3, 361–371. MR 703083, DOI 10.1016/0022-1236(83)90018-6
- Tadashi Huruya, A spectral characterization of a class of $C^*$-algebras, Sci. Rep. Niigata Univ. Ser. A 15 (1978), 21–24. MR 482243
- Helge Elbrønd Jensen, Scattered $C^*$-algebras, Math. Scand. 41 (1977), no. 2, 308–314. MR 482242, DOI 10.7146/math.scand.a-11723
- Masahiro Nakamura, On the direct product of finite factors, Tohoku Math. J. (2) 6 (1954), 205–207. MR 70065, DOI 10.2748/tmj/1178245180
- A. Pełczyński and Z. Semadeni, Spaces of continuous functions. III. Spaces $C(\Omega )$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222. MR 107806, DOI 10.4064/sm-18-2-211-222
- Masamichi Takesaki, On the direct product of $W^{\ast }$-factors, Tohoku Math. J. (2) 10 (1958), 116–119. MR 100798, DOI 10.2748/tmj/1178244749
- Takasi Turumaru, On the direct product of operator algebras. IV, Tohoku Math. J. (2) 8 (1956), 281–285. MR 91435, DOI 10.2748/tmj/1178244952
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 666-668
- MSC: Primary 46L05; Secondary 46M05
- DOI: https://doi.org/10.1090/S0002-9939-1987-0894435-4
- MathSciNet review: 894435