The Commutation Theorem for Tensor Products over von Neumann Algebras

✍ Scribed by Şerban Strătilă; László Zsidó

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 340 KB
• Volume: 165
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3408

✦ Synopsis

dedicated to professor marc a. rieffel on the occasion of his sixtieth birthday A general commutation theorem is proved for tensor products of von Neumann algebras over common von Neumann subalgebras. Roughly speaking, if the noncommon parts of two von Neumann algebras M 1 and M 2 on the same Hilbert space are appropriately separated by commuting type I von Neumann algebras N 1 and N 2 , then the commutant of the von Neumann algebra generated by M 1 and M 2 is generated by the relative commutants M$ 1 & N 1 and M$ 2 & N 2 , as well as by the intersection of the commutants of all concerned von Neumann algebras. This theorem extends both Tomita's classical commutation theorem and a splitting result in tensor products, proved recently in the factor case by L. Ge and R. V. Kadison. Applications are given to a decomposition criterion in ordinary tensor products and to a partial solution of a conjecture of S. Popa concerning the maximal injectivity of tensor products of maximal injective von Neumann subalgebras.