## The slice map problem for $\sigma$-weakly closed subspaces of von Neumann algebras

HTML articles powered by AMS MathViewer

- 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, - 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, - 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
—, - 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

*Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups*, preprint.

*Reflexivity and tensor-products for operator algebras and subspace lattices*, preprint.

*Tensor products and projections of norm one in von Neumann algebras*, Lecture Notes, University of Copenhagen, 1970.

## 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