Compactness properties of operator multipliers.pdf

a r X i v : 0 8 0 9 .2 1 5 8 v 1 [ m a t h .O A ] 1 2 S e p 2 0 0 8 Compactness properties of operator multipliers K. Juschenko, R. H. Levene, I. G. Todorov and L. Turowska September 12, 2008 Abstract We continue the study of multidimensional operator multipliers initiated in [12]. 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 [21]. 1 Introduction A bounded function ? : N×N → C is called a Schur multiplier if (?(i, j)aij) is the matrix of a bounded linear operator on ?2 whenever (aij) is such. The study of Schur multipliers was initiated by Schur in the early 20th century and since then has attracted considerable attention, much of which was inspired by A. Grothendieck’s characterisation of these objects in his Re?sume? [9]. Grothendieck showed that a function ? is a Schur multiplier precisely when it has the form ?(i, j) = ∑∞ k=1 ak(i)bk(j), where ak, bk : N → C satisfy the conditions supi ∑∞ k=1 |ak(i)| 2 ∞ and supj ∑∞ k=1 |bk(j)| 2 ∞. In modern terminology, this characterisation can be expressed by saying that ? is a 0Primary: 46L06, Secondary: 46L07, 47L25 0Keywords: operator multiplier, complete compactness, Schur multiplier, Haagerup tensor product 1 Schur multiplier precisely when it belongs to the extended Haagerup tensor product ?∞ ?eh ?∞ of two copies of ?∞. Special classes of Schur multipliers, e.g. Toeplitz and Hankel Sc