Some Beurling–Fourier algebras on compact groups are operator algebras

Ghandehari, Mahya; Lee, Hun Hee; Samei, Ebrahim; Spronk, Nico

doi:10.1090/tran6653

Some Beurling–Fourier algebras on compact groups are operator algebras

Authors: Mahya Ghandehari, Hun Hee Lee, Ebrahim Samei and Nico Spronk
Journal: Trans. Amer. Math. Soc. 367 (2015), 7029-7059
MSC (2010): Primary 43A30, 47L30, 47L25; Secondary 43A75
DOI: https://doi.org/10.1090/tran6653
Published electronically: April 20, 2015
MathSciNet review: 3378823
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Abstract: Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than half of the dimension of the group. The case of $SU(n)$ will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

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