Almost periodic functionals and finite-dimensional representations

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>We show that if \(A\) is a \(\mathrm{C}^*\)-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.</p></div>