Grant: Discovery Grants Program - Accelerator Supplements

My research explores the fruitful interactions between functional analysis and abstract harmonic analysis, which naturally take place in the framework of Banach and operator algebras. The latter field originated historically from quantum mechanics, and has since then maintained close connections to this and other areas of mathematical physics. One of the most recent developments is the theory of operator spaces and quantum groups, which provides an exciting blend of functional analysis, harmonic analysis, algebra and topology, and has applications to quantum information theory and quantum computing. The proposal mainly aims at thoroughly investigating the structure of certain natural Banach algebras arising in abstract harmonic analysis, i.e., algebras over locally compact groups, as well as their 'quantized' analogues, defined over quantum groups. The key idea is to follow a representation theoretical approach: the objects of interest can be viewed as completely bounded maps on the algebra of bounded linear operators on a suitable Hilbert space. This enables us to use the powerful machinery of von Neumann algebras and operator spaces. In recent joint work with M. Junge and Z.-J. Ruan, we have thus succeeded, for instance, in characterizing one of the most fundamental concepts of Fourier analysis - Pontryagin duality - in terms of a commutation relation, even in the general context of locally compact quantum groups. Moreover, we have been led to exciting discoveries in quantum information theory, namely the construction of new quantum channels with highly desirable properties (in particular, in the setting of finite-dimensional quantum groups, the completely bounded entropy of these channels can be calculated explicitly). One of the main goals pursued in this line of research is the construction of a certain locally compact group associated with every locally compact quantum group, which is an invariant of the latter, and allows us to transfer ideas from classical harmonic analysis to the realm of quantum groups - a fascinating program that I have just started to develop with one of my Ph.D. students, M. Kalantar.