Mappings Preserving Submodules of Hilbert C*‐Modules

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 ∥:=∥〈 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 A X the algebra of all mappings on X of the form 1.1( x ) = ∑ i ‐ 1 ma i x b i   x ∈ Xwhere m is an integer and a i ∈ A , b i ∈ 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.