Bounded functions on the character variety

This paper is motivated by an open question in $p$-adic Fourier theory, that seems to be more difficult than it appears at first glance. Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $o_L$ and let $\mathbb{C}_p$ denote the completion of an algebraic closure of $\mathbb{Q}_p$. In their work on $p$-adic Fourier theory, Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$. This character variety is a rigid analytic curve over $L$ that parameterizes the set of locally $L$-analytic characters $\lambda : (o_L,+) \to (\mathbb{C}_p^\times,\times)$. One of the main results of Schneider and Teitelbaum is that over $\mathbb{C}_p$, the curve $\mathfrak{X}$ becomes isomorphic to the open unit disk. Let $\Lambda_L(\mathfrak{X})$ denote the ring of bounded-by-one functions on $\mathfrak{X}$. If $\mu \in o_L [\![o_L]\!]$ is a measure on $o_L$, then $\lambda \mapsto \mu(\lambda)$ gives rise to an element of $\Lambda_L(\mathfrak{X})$. The resulting map $o_L [\![o_L]\!] \to \Lambda_L(\mathfrak{X})$ is injective. The question is: do we have $\Lambda_L(\mathfrak{X}) = o_L [\![o_L]\!]$? In this paper, we prove various results that were obtained while studying this question. In particular, we give several criteria for a positive answer to the above question. We also recall and prove the ``Katz isomorphism'' that describes the dual of a certain space of continuous functions on $o_L$. An important part of our paper is devoted to providing a proof of this theorem which was stated in 1977 by Katz. We then show how it applies to the question. Besides $p$-adic Fourier theory, the above question is related to the theory of formal groups, the theory of integer valued polynomials on $o_L$, $p$-adic Hodge theory, and Iwasawa theory.