Operator Algebras, Operator Theory and Free Probability Investigations

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Operators on Hilbert space provide the infinite-dimensional linear algebra framework for quantum mechanical observables. Since composition of two linear maps depends on which of the maps acts first, this is a noncommutative mathematics setting where noncommutative analogues of basic mathematical theories are being developed. The principal investigator will work in two directions involving Hilbert space operators, following recent realizations in his research. On one hand the exploration of invariant properties of n-tuples of operators under certain types of perturbations leads to the study of the rings of operators which almost commute in a certain sense with the n-tuple. Connections to operator theory questions as well as to noncommutative geometry in the sense of Alain Connes are appearing. The second direction is in the area of free probability, the most noncommutative among noncommutative probability settings. Free probability, which was initiated by the principal investigator, has important connections to random matrices, combinatorics and to operator algebras. Via the random matrix connection, there are applications to certain models in physics and to models of multiuser communication systems. Recently the principal investigator found that free probability has an extension to a so-called bifree probability theory, which is a fast developing new area, with many open problems and which will also be part of the free probability problems studied. In more detail, the principal investigator will explore on one hand commutants modulo normed ideals of operators, smaller than the ideal of compact operators, which constitute a new operator algebra framework for the study of normed ideal perturbations of operators and on the other hand the bifree extension of free probability, to systems with noncommuting left and right variables. The operator theory questions involve obstructions to quasicentral approximate units relative to a normed ideal with diverse relations to almost normal operators, entropy and supramenable groups. The commutants modulo normed ideals are Banach algebras with involution of operators on Hilbert space, which have unexpectedly many relations to C*-algebras, like in the appearance of C*-algebras which are coronas of non-C*-Banach-algebras. The K-theory aspects of these algebras are also of interest, for instance they provide a new angle on results about invariance of absolutely continuous spectra under perturbations . On the side of free probability and of its bifree extension the emphasis will be on the analysis, while other researchers emphasize the combinatorics. Trace-class commutators of hermitian operators appear both in questions about commutants modulo normed ideals and in free and bifree probability, with the possibility of being a point where the two subjects meet. The principal investigator expects to attract young mathematicians to research in areas of his investigations and in particular to continue organizing conferences and seminars about free probability, which has many connections to other fields in mathematics and provides good training for young mathematicians.