Representations of completely bounded multilinear operators.

*(English)*Zbl 0622.46040There 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 [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 [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. V. I. Paulsen and 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 [E. Christensen, E. G. Effros and 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).

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 [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. V. I. Paulsen and 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 [E. Christensen, E. G. Effros and 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).

##### MSC:

46L05 | General theory of \(C^*\)-algebras |

##### Keywords:

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
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\textit{E. Christensen} and \textit{A. M. Sinclair}, J. Funct. Anal. 72, 151--181 (1987; Zbl 0622.46040)

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##### References:

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