Linear preservers arising from Schur products

Despite this analogous result to that obtained for Schur multiplicative contractions was derived. Results relying on both the Haagerup-Paulsen-Wittstock for completely bounded maps on the bounded operators on the square integrable sequences, and the Stinespring theorem for completely positive maps were obtained. Equivalent forms were obtained for the set of Schur multipliers on this algebra. <br><br>A Schur product for matrix-like tensor product spaces was established and forms for both Schur null preserving and Schur multiplicative maps were obtained. <br><br>Characterisations for even more generalised algebras were sought in relation to hyperfinite factors relying extensively on the Feldman-Moore coordinatisation (as this provides an explicit way to construct the algebra from a sequence of embedded matrices in some cases). <br><br>This proved unsuccessful despite the ability to define a Schur product due to the inability to ensure Schur nullity was preserved by the conditional expectation onto finite dimensional approximating matrix algebras. <br><br>Similar problems and those of even defining Schur products were encountered for more general hyperfinite, uniformly hyperfinite and almost finite algebras; however with the case of compact operators on the square-integrable sequences, a specialised case of an AF algebra, characterisations of both Schur null preservers and Schur multiplicative maps were obtained.