  In this paper we generalize work of Amice and Lazard from the early
(nineteen) sixties. Amice determined the dual of the space of locally
Qp-analytic functions on Zp and showed that it is isomorphic to the ring of
rigid functions on the open unit disk over Cp. Lazard showed that this ring has
a divisor theory and that the classes of closed, finitely generated, and
principal ideals in this ring coincide.
  We study the space of locally L-analytic functions on the ring of integers in
L, where L is a finite extension of Qp. We show that the dual of this space is
a ring isomorphic to the ring of rigid functions on a certain rigid variety X.
We show that the variety X is isomorphic to the open unit disk over Cp, but not
over any discretely valued extension field of L; it is a "twisted form" of the
open unit disk. In the ring of functions on X, the classes of closed, finitely
generated, and invertible ideals coincide, but unless L=Qp not all finitely
generated ideals are principal.
  The paper uses Lubin-Tate theory and results on p-adic Hodge theory. We give
several applications, including one to the construction of p-adic L-functions
for supersingular elliptic curves.
