A Factorization Problem for Normal Completely Bounded Mappings

✍ Scribed by Christian Le Merdy; Bojan Magajna

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 245 KB
• Volume: 181
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3715

✦ Synopsis

Given an operator space X and a von Neumann algebra A, we consider a contractive mapping q: A eh X eh A Ä NCB(X*, A) formally defined by q( a j x jk b k )= x jk a j b k , from the extended Haagerup tensor product A eh X eh A into the space of w*-continuous completely bounded maps from X* into A. We characterize elements of the range space Im(q) by a factorization property involving decomposable operators and investigate various properties of that space. In the case when X=B * is the predual of a von Neumann algebra B, Im(q) is included in the space DEC(B, A) of decomposable operators from B into A. Regarding q as having values in that space, we show that q is a quotient map onto its range. Then we prove that DEC(B, A) is a normal dual operator A-bimodule and that Im(q)/ DEC(B, A) is a strong operator A-bimodule.