  A-manifolds and A-bundles are manifolds and vector bundles modelled on a
projective finitely generated module over a topological algebra A. In this
paper we investigate the conditions under which an A-bundle is provided with an
A-valued hermitian structure and a compatible connection, in case A is a
commutative complete locally m-convex C*-algebra with unit. In this
investigation two obstacles appear: first, A-manifolds do not admit partitions
of unity in the classical sence, and, secondly, the existence of a hermitian
structure is not equivalent to the reduction of the structural group of the
bundle to a certain subgroup. However, we prove that if the bundle has a
trivializing covering whose transition functions take values in the group of
the "A-hermitian product preserving" automorphisms of the fibre type and the
base space admits at least one A-valued partition of unity (subordinate to this
covering), then the A-bundle admits an A-hermitian structure and a compatible
connection.
