Asymptotic abelianness of infinite factors

Glaser, M. S.

doi:10.1090/S0002-9947-1973-0317062-0

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

Les algébres d’opérateurs dans l’espace hilbertien

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

The theory of

algebras

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