Almost periodic functionals and finite-dimensional representations
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.