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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multidimensional operator multipliers


Authors: K. Juschenko, I. G. Todorov and L. Turowska
Journal: Trans. Amer. Math. Soc. 361 (2009), 4683-4720
MSC (2000): Primary 46L07; Secondary 47L25
Published electronically: April 10, 2009
MathSciNet review: 2506424
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Abstract: We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several $ C^*$-algebras satisfying certain boundedness conditions. In the case of commutative $ C^*$-algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding $ C^*$-algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding $ C^*$-algebras.


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Additional Information

K. Juschenko
Affiliation: Department of Mathematics, Chalmers Institute of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: jushenko@chalmers.se

I. G. Todorov
Affiliation: Department of Pure Mathematics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom
Email: i.todorov@qub.ac.uk

L. Turowska
Affiliation: Department of Mathematics, Chalmers Institute of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: turowska@chalmers.se

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04771-0
PII: S 0002-9947(09)04771-0
Keywords: Multiplier, $C^*$-algebra, multidimensional
Received by editor(s): July 5, 2007
Published electronically: April 10, 2009
Article copyright: © Copyright 2009 American Mathematical Society