A Hilbert module over a C*-algebra B is a right
B-module X, equipped with an
inner product 〈·, ·〉 which is linear over
B in the second factor, such that X is a Banach space
with the norm
∥x∥[ratio ]=∥〈x, x〉∥1/2.
(We refer to  for the basic theory
of Hilbert modules; the basic example for us will be X=B
with the inner product 〈x, y〉=x*y.)
We denote by B(X) the
algebra of all bounded linear operators on X,
and we denote by L(X) the C*-algebra
of all adjointable operators. (In the basic
example X=B, L(X) is just the multiplier
algebra of B.) Let A be a C*-subalgebra
of L(X), so that X is an A-B-bimodule.
We always assume that A is nondegenerate in
the sense that [AX]=X, where [AX]
denotes the closed linear span of AX.
Denote by AX the algebra of all mappings
on X of the form
where m is an integer and ai∈A,
bi∈B for all i.
Mappings of form (1.1) will be called
elementary, and this paper is concerned with the
question of which mappings on X
can be approximated by elementary mappings in the point norm topology.