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 [8] 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

formula here

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.