Thebookfeaturesnewdirectionsinanalysis,withanemphasisonHilbertspace,mathematicalphysics,andstochasticprocesses.Weinterpret'non-commutativeanalysis'broadlytoincluderepresentationsofnon-Abeliangroups,andnon-Abelianalgebras;emphasisonLiegroupsandoperatoralgebras(C*algebrasandvonNeumannalgebras.)Asecondthemeiscommutativeandnon-commutativeharmonicanalysis,spectraltheory,operatortheoryandtheirapplications.Thelistoftopicsincludesshiftinvariantspaces,groupactionindifferentialgeometry,andframetheory(over-completebases)andtheirapplicationstoengineering(signalprocessingandmultiplexing),projectivemulti-resolutions,andfreeprobabilityalgebras.Thebookservesasanaccessibleintroduction,offeringatimelesspresentation,attractiveandaccessibletostudents,bothinmathematicsandinneighboringfields.