Uncertainty principles for Kac algebras

Abstract

In this paper, we introduce the notation of bi-shift of biprojections in subfactor theory to unimodular Kac algebras. We characterize the minimizers of Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for unimodular Kac algebras with biprojections and prove Hardy's uncertainty principle in terms of minimizers. 1 Introduction Uncertainty principles for locally compact abelian groups were studied by Hardy [15], Hirschman [16], Beckner [2], Donoho and Stark [9], Smith [23], Tao [24] etc. In 2008, Alagic and Russell [1] proved Donoho-Stark uncertainty principle for compact groups. In 2004,Özaydm and Przebinda [21] characterized the minimizers of Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for locally compact abelian groups.Kac algebras were introduced independently by L.I Vainerman and G.I. Kac [27,28,29] and by Enock and Nest [10,11,12], which generalized locally compact groups and their duals. Furthermore, J. Kustermans and S. Vaes introduced locally compact quantum groups [18]. Recently Crann and Kalantar proved Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle for unimodular locally compact quantum groups [7].Subfactor theory also provides a natural framework to study quantum symmetry. The group symmetry is captured by the subfactor arisen from the group crossed product construction. Ocneanu first pointed out the one-to-one correspondence between finite dimensional Kac algebras and finiteindex, depth-two, irreducible subfactors. This correspondence was proved by W. Szymanski [22]. Enock and Nest generalized the correspondence to infinite dimensional compact (or discrete) type Kac algebras and infinite-index, depth-two, irreducible subfactors [14]. In general, a subfactor provides a pair of non-commutative spaces dual to each other and a Fourier transform F between them. It appears to be natural to study Fourier analysis for subfactors.

Quantum Fourier analysis

“…The quantum inequalities in Theorem 2.1 on these infinite quantum symmetries have been partially studied in refs. [36][37][38][39][40]. The quantum uncertainty principle QUP -2 in Theorem 2.1 becomes a continuous family of inequalities on locally compact quantum groups (40).…”

Quantum smooth uncertainty principles for von Neumann bi-algebras

“…Recently quantum uncertainty principles on subfactors, an important type of quantum symmetires [11,5], have been established for support and for von Neumann entropy in [9] and for Rényi entropy in [18]. These quantum uncertainty principles have been generalized on other types of quantum symmetries, such as Kac algebras [17], locally compact quantum groups [10] and fusion bialgebras [16] etc, in the unified framework of quantum Fourier analysis [8]. Such quantum inequalities were applied in the classification of subfactors [15] and as analytic obstructions of unitary categorifications of fusion rings in [16].…”

Galois Correspondence and Fourier Analysis on Local Discrete Subfactors

“…Since then, it has been a cornerstone in the analysis of subfactors. More recently, is has been extensively studied for finite index subfactors and planar algebras [JLW16], for Kac algebras [LW17] and locally compact quantum groups [JLW18], proving a number of inequalities and uncertainty principles which generalize classical results from the Fourier analysis on groups. See [JJL + 20] for a concise description of the program.…”