Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the size of orbits in the duals of $ C^*$-algebras and convolution algebras


Author: Matthias Neufang
Journal: Proc. Amer. Math. Soc. 148 (2020), 667-671
MSC (2010): Primary 43A20, 46H25, 46J10, 46J40, 46L05
DOI: https://doi.org/10.1090/proc/14780
Published electronically: October 18, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We solve an open problem raised in [Trans. Amer. Math. Soc. 366 (2014), pp.4151-4171] concerning (infinite-dimensional) commutative semisimple Banach algebras $ \mathcal {A}$ with a bounded approximate identity, namely, whether there always exists a functional $ f \in \mathcal {A}^*$ such that the orbit subspace $ \mathcal {A}^{**} \Box f$ of $ \mathcal {A}^*$ is $ w^*$-closed and infinite-dimensional. Indeed, on the one hand, we show that no commutative $ C^*$-algebra shares this property. On the other hand, we prove that the answer is positive, in a strong sense, in the case of convolution algebras $ \mathcal {A} = L_1(\mathcal {G})$, for large classes of locally compact groups $ \mathcal {G}$ (commutativity is not needed): there exists $ f \in \mathcal {A}^*$ with maximal orbit, in fact, $ f$ satisfies $ \mathrm {Ball} (\mathcal {A}^*) = \mathrm {Ball}(\mathcal {A}^{**}) \Box f$. Moreover, as we shall see, the latter property links the size of orbits to Arens irregularity.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 43A20, 46H25, 46J10, 46J40, 46L05

Retrieve articles in all journals with MSC (2010): 43A20, 46H25, 46J10, 46J40, 46L05


Additional Information

Matthias Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada; and Laboratoire de Mathématiques Paul Painlevé (UMR CNRS 8524), Université de Lille, UFR de Mathématiques, 59655 Villeneuve d’Ascq Cedex, France
Email: mneufang@math.carleton.ca; matthias.neufang@univ-lille.fr

DOI: https://doi.org/10.1090/proc/14780
Received by editor(s): May 6, 2019
Published electronically: October 18, 2019
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2019 American Mathematical Society