This book offers a presentation of some new trends in operator theory and operator algebras, with a view to their applications. It consists of separate papers written by some of the leading practitioners in the field. The content is put together by the three editors in a way that should help students and working mathematicians in other parts of the mathematical sciences gain insight into an important part of modern mathematics and its applications. While different specialist authors are outlining new results in this book, the presentations have been made user friendly with the aid of tutorial material.In fact, each paper contains three things: a friendly introduction with motivation, tutorial material, and new research. The authors have strived to make their results relevant to the rest of mathematics. A list of topics discussed in the book includes wavelets, frames and their applications, quantum dynamics, multi variable operator theory, $C^*$-algebras, and von Neumann algebras. Some longer papers present recent advances on particular, long-standing problems such as extensions and dilations, the Kadison-Singer conjecture, and diagonals of self-adjoint operators.

Table of Contents:
Applications, wavelets, dynamics, and quantum theory: Weyl-Heisenberg frame wavelets with basic supports by X. Guo, Y. Diao, and X. Dai Use of operator algebras in the analysis of measures from wavelets and iterated function systems by P. E. T. Jorgensen A brief introduction to operator quantum error correction by D. W. Kribs From filters to wavelets via direct limits by N. S. Larsen and I. Raebum Wavelet image compression by M.-S. Song Operator theory and harmonic analysis: Wiener-Hopf plus Hankel operators on the real line with unitary and sectorial symbols by G. Bogveradze and L. P. Castro An explicit duality for finite groups by R. A. Cohen and M. E. Walter The corona factorization property by P. W. Ng Factorization theory for Wiener-Hopf plus Hankel operators with almost periodic symbols by A. P. Nolasco and L. P. Castro Non-commutative geometry: On spectral invariance of non-commutative tori by F. Luef Lipschitz extension constants equal projection constants by M. A. Rieffel $C^*$-algebras: Certain representations of the Cuntz relations, and a question on wavelets decompositions by P. E. T. Jorgensen The Rokhlin property for automorphisms on simple $C^*$-algebras by H. Lin Some recent topics on $C^*$-algebras and related problems by S. Sakai Strong shift equivalence in the $C^*$-algebraic setting: Graphs and $C^*$-correspondences by M. Tomforde Topological higher-rank graphs and the $C^*$-algebras of topological 1-graphs by T. Yeend von Neumann algebras: Diagonals of self-adjoint operators by W. Arveson and R. V. Kadison On the Laplacian subalgebra of certain tensors of group von Neumann algebras by T. S. Bildea Frames in Hilbert space: Structured Parseval frames in Hilbert $C^*$-modules by W. Jing, D. Han, and R. N. Mohapatra Robertson-type theorems for countable groups of unitary operators by D. R. Larson, W.-S. Tang, and E. Weber The Kadison-Singer conjecture: The Kadison-Singer problem in mathematics and engineering: A detailed account by P. G. Casazza, M. Fickus, J. C. Tremain, and E. Weber Extensions and dilations: The gauge group of an $E_0$-semigroup by A. Alevras Extensions and dilations for $C^*$-dynamical systems by P. S. Muhly and B. Solel On some $E_0$-semigroups induced from $CP$-flows on $\mathfrak{B}(\mathfrak{h})$ by R. T. Powers and G. L. Price Multivariable operator theory: Trace-class perturbations and functional calculus by H. Bercovici and D. Timotin Rank preserving maps on CSL algebras by J. Kim and R. L. Moore.