Multipliers and the Similarity Property for Topological Quantum Groups

Abstract

Studying Lp-space and Fourier analysis on Euclidean space is one of the main interests in analysis, and a number of themes emerged from harmonic analysis, probability and partial differential equations are being actively explored. Other than the Euclidean space, various studies have been conducted on Lie groups, more generally on topological groups, which have become the backbone of abstract harmonic analysis.<br/> The theory of operator algebras has contributed to understand topological quantum groups. Through the pioneering works [Wor87b], [Wor87a], [KV00] and [KV03], theories of locally compact quantum groups have been successfully developed. On the other hand, functional analysis provides us with an environment in which non-commutative Lp-analysis can be explored. In particular, Fourier analysis has been actively studied on quantum tori or quantum groups.<br/> Throughout this thesis, we will synthesize the results of [You18b], [BY17], [You17] and [You18a], which have been studied by the author during the doctorate course. Main results can be summarized as follows:<br/> - For a compact group G, it is known that all random Fourier series of f in M(G) are in M(G) if and only if f is in L2(G) [Hel57]. We show that this result still holds for compact quantum groups of Kac type, which characterizes all bounded multipliers from Lp into lq for all 2<= p and 1<= q<= 2. We provide some partial results for non-Kac cases and apply these results to a representability problem on convolution algebras of compact quantum groups as operator algebras [You17].<br/> - For compact quantum groups of Kac type, we establish Hardy-Littlewood inequalities through geometric information of the underlying quantum group and prove that the obtained inequalities are sharp for important examples [You18b]. These inequalities can be interpreted as bounded multipliers from Lp into lp with 1< p< 2.<br/> - Uncertainty principle implies that both non-zero function and its Fourier transform cannot be sharply localized. This principle does hold for locally compact quantum groups ([CK14], [LW17] and [JLW18]), and we prove that f is much more dispersed than expected if its Fourier transform is localized on certain discrete quantum groups. Moreover, we explain that this is an exceedingly unique phenomenon [You18a].<br/> - While the amenability implies unitarizability in the realm of topological groups, we show that the natural analogue does not hold for quantum groups. More precisely, it is known that unitarizability holds if we impose an additional assumption, and we show that this assumption is indispensable. Also, we prove that the assumption can be removed if the quantum group is amenable, discrete and of Kac type [BY17].<br/> <br/>