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Almost periodic functionals and finite-dimensional representations

Volume 243 / 2018

M. Filali, M. Sangani Monfared Studia Mathematica 243 (2018), 329-344 MSC: 46H15, 43A60, 43A20, 22D10, 22D20. DOI: 10.4064/sm170626-26-9 Published online: 7 May 2018

Abstract

We show that if $A$ is a C$^*$-algebra and $\lambda \in A^*$ is a nonzero almost periodic functional which is a coordinate functional of a topologically irreducible involutive representation $\pi $, then $\dim\pi \lt \infty $. We introduce the RFD transform $\alpha _A : A \rightarrow U(A)$ of a Banach algebra $A$ and establish its universal property. We show that if $A$ has a bounded two-sided approximate identity, then almost periodic functionals on $A$ which are limits of coordinate functionals of finite-dimensional representations have lifts to almost periodic functionals on $U(A)$. Other connections with almost periodicity and harmonic analysis are also discussed.

Authors

  • M. FilaliDepartment of Mathematical Sciences
    University of Oulu
    Oulu 90014, Finland
    e-mail
  • M. Sangani MonfaredDepartment of Mathematics and Statistics
    University of Windsor
    Windsor, ON, N9B 3P4, Canada
    e-mail

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