an:04009102
Zbl 0622.46040
Christensen, Erik; Sinclair, Allan M.
Representations of completely bounded multilinear operators
EN
J. Funct. Anal. 72, 151-181 (1987).
00153291
1987
j
46L05
representation theorem; completely bounded linear operator from a \(C^ *\)-algebra into the algebra of bounded linear operators; complete boundedness for a multilinear operator from one \(C^ *\)-algebra; Wittstock's matricial Hahn-Banach Theorem; injective von Neumann algebra
There is a well known representation theorem for a completely bounded linear operator from a \(C^*\)-algebra into the algebra of bounded linear operators BL(H) on a Hilbert space H. This result may be proved either by employing Wittstock's decomposition of a completely bounded linear operator into a linear combination of four completely positive operators or by Arveson's extension theorem together with a dilation argument. See [\textit{V. I. Paulsen}, Completely bounded maps and dilations, Pitman Research Notes Math., 146 (1986; Zbl 0614.47006)] for a detailed discussion of completely bounded linear operators.
In this paper the authors introduce a definition of complete boundedness for a multilinear operator from one \(C^*\)-algebra into another. Using Wittstock's matricial Hahn-Banach Theorem [\textit{G. Wittstock}, J. Funct. Anal. 40, 127-150 (1981; Zbl 0495.46005)] they prove a corresponding representation theorem for completely bounded multilinear operators in terms of *-representations of the algebra and suitable bridging operators between the Hilbert spaces. \textit{V. I. Paulsen} and \textit{R. R. Smith} have recently given a proof of this result that depends on Arveson's extension theorem, and have extended the result to operator spaces [J. Funct. Anal. 73, 258-276 (1987)]. The representation theorem has been used to show that certain cohomology groups from a von Neumann algebra into an injective von Neumann algebra are zero [\textit{E. Christensen, E. G. Effros} and \textit{A. M. Sinclair}, Inventiones Math. 90, 279-296 (1987)].
(Correction: E. G. Effros has pointed out that \(K^{\infty}\) in Corollaries 5.7 and 5.8 should be replaced by \(K^ I=K\otimes \ell^ 2(I)\) for an arbitrary set I. Alternatively the conclusions are correct with the additional assumptions that the Hilbert spaces are separable and the von Neumann algebras have separable preduals).
Zbl 0614.47006; Zbl 0495.46005