### Abstract

Let A(1), A(2),..., A(r) be C*-algebras with second duals A"(1), A"(2),...,A"(r), and let X be an arbitrary trary Banach space. Let Gamma: A"(1) x A"(2) x (...) x A"(r) --> X" be a bounded r-linear map, and denote by Gamma": A"(1) x A"(2) x (...) x A"(r) --> X" the Johnson-Kadison-Rinrose extension (i.e., the separately weak* to weak* continuous r-linear extension) of Gamma. The problem of characterising those Gamma for which Gamma" takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C*-algebras, the 'obvious' extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C*-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra. (C) 2004 Elsevier Inc. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 558-570 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 292 |

DOIs | |

Publication status | Published - 2004 |

### Keywords

- DUNFORD-PETTIS PROPERTY

### Cite this

*Journal of Mathematical Analysis and Applications*,

*292*, 558-570. https://doi.org/10.1016/j.jmaa.2003.12.024

**Multilinear maps on products of operator algebras.** / Wright, John David Maitland; Ylinen, K.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 292, pp. 558-570. https://doi.org/10.1016/j.jmaa.2003.12.024

}

TY - JOUR

T1 - Multilinear maps on products of operator algebras

AU - Wright, John David Maitland

AU - Ylinen, K.

PY - 2004

Y1 - 2004

N2 - Let A(1), A(2),..., A(r) be C*-algebras with second duals A"(1), A"(2),...,A"(r), and let X be an arbitrary trary Banach space. Let Gamma: A"(1) x A"(2) x (...) x A"(r) --> X" be a bounded r-linear map, and denote by Gamma": A"(1) x A"(2) x (...) x A"(r) --> X" the Johnson-Kadison-Rinrose extension (i.e., the separately weak* to weak* continuous r-linear extension) of Gamma. The problem of characterising those Gamma for which Gamma" takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C*-algebras, the 'obvious' extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C*-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra. (C) 2004 Elsevier Inc. All rights reserved.

AB - Let A(1), A(2),..., A(r) be C*-algebras with second duals A"(1), A"(2),...,A"(r), and let X be an arbitrary trary Banach space. Let Gamma: A"(1) x A"(2) x (...) x A"(r) --> X" be a bounded r-linear map, and denote by Gamma": A"(1) x A"(2) x (...) x A"(r) --> X" the Johnson-Kadison-Rinrose extension (i.e., the separately weak* to weak* continuous r-linear extension) of Gamma. The problem of characterising those Gamma for which Gamma" takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C*-algebras, the 'obvious' extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C*-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra. (C) 2004 Elsevier Inc. All rights reserved.

KW - DUNFORD-PETTIS PROPERTY

U2 - 10.1016/j.jmaa.2003.12.024

DO - 10.1016/j.jmaa.2003.12.024

M3 - Article

VL - 292

SP - 558

EP - 570

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

ER -