Unbounded generalisations of Schur and operator multipliers
Naomi M. Steen · OpenGrey (Institut de l'Information Scientifique et Technique) · 2013
Bounded Schur multipliers were introduced and characterised several decades ago, and various applications of this algebra of functions have been discovered. More recently, research into different classes of unbounded multipliers has been carried out. In this thesis the theory of one such class, that of the local Schur multipliers, is extended in different settings. A dilation of minimal Stinespring representations of completely positive, bimodular maps on spaces of compact operators is obtained, and used to establish an unbounded version of Stinespring's Theorem. This theorem is applied to obtain a characterisation of positive local Schur multipliers. In addition, a relation is demonstrated between operator monotone functions and positive local Schur multipliers, and a description is given of positive multipliers of Toeplitz type. The theory of local multipliers is extended to the multidimensional setting, and a characterisation of such functions is obtained. Local operator multipliers are introduced as a non-commutative e analogue of local Schur multipliers and a description is provided, extending previously known results concerning completely bounded operator multipliers. Positive multipliers are defined in this setting) and characterised using elements of canonical positive cones. The two-dimensional Fourier algebra A2(G) of a compact, abelian group G is considered, and a number of results are obtained concerning the Arens product on its dual, VN(G) ®uh VN(G). It is shown that A2 (G) may be viewed as a left VN(G) ®ub VN(G)-module, and thus certain results of Eyroard are extended to the two-dimensional setting, leading to the establishment of a condition equivalent to the homeomorphic identification of the Gelfand spectrum of A2(G) with C2 .