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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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Additional Information

Mahya Ghandehari
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Email: m.ghandehary@gmail.com

Hun Hee Lee
Affiliation: Department of Mathematical Sciences, Seoul National University, Gwanak-ro 1 Gwanak-gu, Seoul 151-747, Republic of Korea
Email: hunheelee@snu.ac.kr

Ebrahim Samei
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada
Email: samei@math.usask.ca

Nico Spronk
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Email: nspronk@math.uwaterloo.ca

DOI: https://doi.org/10.1090/tran6653
Keywords: Weighted Fourier algebras, operator algebras, compact connected Lie groups, Littlewood multipliers
Received by editor(s): July 22, 2013
Published electronically: April 20, 2015
Additional Notes: The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2005963)
The third author was supported by NSERC under grant no. 366066-09
The fourth author was supported by NSERC under grant no. 312515-05
Article copyright: © Copyright 2015 American Mathematical Society