TY - JOUR
T1 - Compactness properties of operator multipliers
AU - Juschenko, K.
AU - Levene, R. H.
AU - Todorov, I. G.
AU - Turowska, L.
N1 - Funding Information:
The first named author was supported by The Royal Swedish Academy of Sciences, Knut och Alice Wallenbergs Stiftelse and Jubileumsfonden of the University of Gothenburg’s Research Foundation. The second and the third named authors were supported by Engineering and Physical Sciences Research Council grant EP/D050677/1. The last named author was supported by the Swedish Research Council.
PY - 2009/6/1
Y1 - 2009/6/1
N2 - We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C*-Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].
AB - We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C*-Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].
KW - Complete compactness
KW - Haagerup tensor product
KW - Operator multiplier
KW - Schur multiplier
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U2 - 10.1016/j.jfa.2008.12.018
DO - 10.1016/j.jfa.2008.12.018
M3 - Article
AN - SCOPUS:64549110682
SN - 0022-1236
VL - 256
SP - 3772
EP - 3805
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 11
ER -