Almost periodic functionals and finite-dimensional representations
Volume 243 / 2018
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.
