The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras
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- by Jon Kraus PDF
- Trans. Amer. Math. Soc. 279 (1983), 357-376 Request permission
Abstract:
A $\sigma$-weakly closed subspace $\mathcal {S}$ of $B(\mathcal {H})$ is said to have Property ${S_\sigma }$ if for any $\sigma$-weakly closed subspace $\mathcal {T}$ of a von Neumann algebra $\mathcal {N},\{ x \in \mathcal {S}\;\overline \otimes \mathcal {N}:{R_\varphi }(x) \in \mathcal {T}\; {\text {for all}}\;\varphi \in B{(\mathcal {H})_{\ast }}\} = \mathcal {S} \overline \otimes \mathcal {T}$, where ${R_\varphi }$ is the right slice map associated with $\varphi$. It is shown that semidiscrete von Neumann algebras have Property ${S_\sigma }$, and various stability properties of the class of $\sigma$-weakly closed subspaces with Property ${S_\sigma }$ are established. It is also shown that if $(\mathcal {M},G,\alpha )$ is a ${W^{\ast }}$-dynamical system such that $\mathcal {M}$ has Property ${S_\sigma }$ and $G$ is compact abelian, then all of the spectral subspaces associated with $\alpha$ have Property ${S_\sigma }$. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if ${\mathcal {L}_1}$ is a commutative subspace lattice with totally atomic core, and ${\mathcal {L}_2}$ is an arbitrary subspace lattice, then ${\text {alg}}({\mathcal {L}_{1}} \otimes {\mathcal {L}_2}) = {\text {alg}}\;{\mathcal {L}_{1}} \overline \otimes {\text {alg}}\;{\mathcal {L}_2}$.References
- William Arveson, On groups of automorphisms of operator algebras, J. Functional Analysis 15 (1974), 217–243. MR 0348518, DOI 10.1016/0022-1236(74)90034-2
- William Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532. MR 365167, DOI 10.2307/1970956
- William Arveson, The harmonic analysis of automorphism groups, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 199–269. MR 679706
- Man Duen Choi and Edward G. Effros, Separable nuclear $C^*$-algebras and injectivity, Duke Math. J. 43 (1976), no. 2, 309–322. MR 405117
- Alain Connes, Une classification des facteurs de type $\textrm {III}$, Ann. Sci. École Norm. Sup. (4) 6 (1973), 133–252 (French). MR 341115
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- A. Connes, On the classification of von Neumann algebras and their automorphisms, Symposia Mathematica, Vol. XX (Convegno sulle Algebre $C^*$ e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria $K$, INDAM, Rome, 1975) Academic Press, London, 1976, pp. 435–478. MR 0450988 J. De Cannière and U. Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, preprint.
- 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
- Frank Gilfeather, Alan Hopenwasser, and David R. Larson, Reflexive algebras with finite width lattices: tensor products, cohomology, compact perturbations, J. Funct. Anal. 55 (1984), no. 2, 176–199. MR 733915, DOI 10.1016/0022-1236(84)90009-0
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- Uffe Haagerup, The reduced $C^{\ast }$-algebra of the free group on two generators, 18th Scandinavian Congress of Mathematicians (Aarhus, 1980) Progr. Math., vol. 11, Birkhäuser, Boston, Mass., 1981, pp. 321–335. MR 633366 K. Harrison, Reflexivity and tensor-products for operator algebras and subspace lattices, preprint.
- Alan Hopenwasser, Cecelia Laurie, and Robert Moore, Reflexive algebras with completely distributive subspace lattices, J. Operator Theory 11 (1984), no. 1, 91–108. MR 739795
- Jon Kraus, $W^{\ast }$-dynamical systems and reflexive operator algebras, J. Operator Theory 8 (1982), no. 1, 181–194. MR 670184
- Richard I. Loebl and Paul S. Muhly, Analyticity and flows in von Neumann algebras, J. Functional Analysis 29 (1978), no. 2, 214–252. MR 504460, DOI 10.1016/0022-1236(78)90007-1
- Dorte Olesen, On spectral subspaces and their applications to automorphism groups, Symposia Mathematica, Vol. XX (Convegno sulle Algebre $C^*$ e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria $K$, INDAM, Rome, 1975) Academic Press, London, 1976, pp. 253–296. MR 0487481
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- Masamichi Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. MR 438149, DOI 10.1007/BF02392041
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- Jun Tomiyama, Tensor products of commutative Banach algebras, Tohoku Math. J. (2) 12 (1960), 147–154. MR 115108, DOI 10.2748/tmj/1178244494 —, Tensor products and projections of norm one in von Neumann algebras, Lecture Notes, University of Copenhagen, 1970.
- Jun Tomiyama, Tensor products and approximation problems of $C^*$-algebras, Publ. Res. Inst. Math. Sci. 11 (1975/76), no. 1, 163–183. MR 0397427, DOI 10.2977/prims/1195191690
- N. Th. Varopoulos, Tensor algebras and harmonic analysis, Acta Math. 119 (1967), 51–112. MR 240564, DOI 10.1007/BF02392079
- 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, On tensor products of certain group $C^{\ast }$-algebras, J. Functional Analysis 23 (1976), no. 3, 239–254. MR 0425628, DOI 10.1016/0022-1236(76)90050-1
- 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
- László Zsidó, On spectral subspaces associated to locally compact abelian groups of operators, Adv. in Math. 36 (1980), no. 3, 213–276. MR 577304, DOI 10.1016/0001-8708(80)90016-X
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 279 (1983), 357-376
- MSC: Primary 46L10; Secondary 46L55, 46M05, 47D25
- DOI: https://doi.org/10.1090/S0002-9947-1983-0704620-0
- MathSciNet review: 704620