  We prove that the Berkovich space of the algebra of bounded analytic
functions on the open unit disk of an algebraically closed nonarchimedean field
contains multiplicative seminorms that are not norms and whose kernel is not a
maximal ideal. We also prove that in general these seminorms are not univocally
determined by their kernels, and provide a method for obtaining families of
different seminorms sharing the same kernel. On the other hand, we prove that
there are also kernels that cannot be obtained by that method. The relation
with the Berkovich space of the Tate algebra is also given.
