## Asymptotic abelianness of infinite factors

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- by M. S. Glaser PDF
- Trans. Amer. Math. Soc.
**178**(1973), 41-56 Request permission

## Abstract:

Studying Pukánszky’s type III factor, ${M_2}$, we show that it does not have the property of asymptotic abelianness and discuss how this property is related to property L. We also prove that there are no asymptotic abelian ${\text {II}_\infty }$ factors. The extension (by ampliation) of central sequences in a finite factor,*N*, to $M \otimes N$ is shown to be central. Also, we give two examples of the reduction (by equivalence) of a central sequence in $M \otimes N$ to a sequence in

*N*. Finally, applying the definition of asymptotic abelianness of ${C^\ast }$-algebras to ${W^\ast }$-algebras leads to the conclusion that all factors satisfying this property are abelian.

## References

- Wai-mee Ching,
*Non-isomorphic non-hyperfinite factors*, Canadian J. Math.**21**(1969), 1293–1308. MR**254614**, DOI 10.4153/CJM-1969-142-6
J. Dixmier, - J. Dixmier,
*Quelques propriétés des suites centrales dans les facteurs de type $\textrm {II}_{1}$*, Invent. Math.**7**(1969), 215–225 (French). MR**248534**, DOI 10.1007/BF01404306 - J. Dixmier and E. C. Lance,
*Deux nouveaux facteurs de type $\textrm {II}_{1}$*, Invent. Math.**7**(1969), 226–234 (French). MR**248535**, DOI 10.1007/BF01404307 - H. A. Dye,
*The unitary structure in finite rings of operators*, Duke Math. J.**20**(1953), 55–69. MR**52695** - Robert R. Kallman,
*One-parameter groups of $^{\ast }$-automorphisms of $II_{1}$ von Neumann algebras*, Proc. Amer. Math. Soc.**24**(1970), 336–340. MR**251548**, DOI 10.1090/S0002-9939-1970-0251548-5 - Robert R. Kallman,
*Unitary groups and automorphisms of operator algebras*, Amer. J. Math.**91**(1969), 785–806. MR**254617**, DOI 10.2307/2373352 - Dusa McDuff,
*Central sequences and the hyperfinite factor*, Proc. London Math. Soc. (3)**21**(1970), 443–461. MR**281018**, DOI 10.1112/plms/s3-21.3.443 - Robert T. Powers,
*Representations of uniformly hyperfinite algebras and their associated von Neumann rings*, Ann. of Math. (2)**86**(1967), 138–171. MR**218905**, DOI 10.2307/1970364 - L. Pukánszky,
*Some examples of factors*, Publ. Math. Debrecen**4**(1956), 135–156. MR**80894** - Shôichirô Sakai,
*Asymptotically abelian $\textrm {II}_{1}$-factors*, Publ. Res. Inst. Math. Sci. Ser. A**4**(1968/1969), 299–307. MR**0248533**, DOI 10.2977/prims/1195194878
—, - J. T. Schwartz,
*$W^{\ast }$-algebras*, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR**0232221** - Jun Tomiyama,
*On the tensor products of von Neumann algebras*, Pacific J. Math.**30**(1969), 263–270. MR**246141**, DOI 10.2140/pjm.1969.30.263 - Paul Willig,
*$B(H)$ is very noncommutative*, Proc. Amer. Math. Soc.**24**(1970), 204–205. MR**248537**, DOI 10.1090/S0002-9939-1970-0248537-3 - Paul Willig,
*Properties $\Gamma$ and $L$ for type $\textrm {II}_{1}$ factors*, Proc. Amer. Math. Soc.**25**(1970), 836–837. MR**259630**, DOI 10.1090/S0002-9939-1970-0259630-3 - G. Zeller-Meier,
*Deux nouveaux facteurs de type $\textrm {II}_{1}$*, Invent. Math.**7**(1969), 235–242 (French). MR**248536**, DOI 10.1007/BF01404308

*Les algébres d’opérateurs dans l’espace hilbertien*, 2nd ed., Gauthier-Villars, Paris, 1969.

*The theory of*${W^\ast }$ -

*algebras*, Yale University, New Haven, Conn., 1962.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 41-56 - MSC: Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0317062-0
- MathSciNet review: 0317062