Schur multipliers of Cartan pairs

We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of B(` 2 ). We characterise these as the normal A-bimodule maps on M. If M contains a direct summand isomorphic to the hyper- finite II1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product A ⊗eh A are strictly contained in the algebra of all Schur multipliers.

Finite domination and Novikov rings. Laurent polynomial rings in several variables

We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded complex of finitely generated free modules). The main tools, which we develop in the paper, are a non-standard totalisation construction for multi-complexes based on truncated products, and a high-dimensional mapping torus construction employing a theory of cubical diagrams that commute up to specified coherent homotopies.

Triangular Objects and Systematic K-Theory

We investigate modules over “systematic” rings. Such rings are “almost graded” and have appeared under various names in the literature; they are special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the presence of conditions on the support, and explain how this generalises and unifies calculations of graded and filtered K-theory scattered in the literature. Our treatment makes systematic use of the formalism of idempotent completion and a theory of triangular objects in additive categories, leading to elementary and transparent proofs throughout.

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

Compressive sensing in Clifford analysis

Compressed sensing is a new paradigm in signal processing which states that for certain matrices sparse representations can be obtained by a simple l1-minimization. In this thesis we explore this paradigm for higher-dimensional signal. In particular three cases are being studied: signals taking values in a bicomplex algebra, quaternionic signals, and complex signals which are representable by a nonlinear Fourier basis, a so-called Takenaka-Malmquist system.