The closure of two-sided multiplications on C*-algebras and phantom line bundles

For a C*-algebra A, we consider the problem when the set TM0(A) of all two-sided multiplications x (sic) axb (a, b is an element of A) on A is norm closed, as a subset of B(A). We first show that TM0(A) is norm closed for all prime C*-algebras A. On the other hand, if A congruent to Gamma(0)(epsilon) is an n-homogeneous C*-algebra, where E is the canonical Mn-bundle over the primitive spectrum X of A, we show that TM0(A) fails to be norm closed if and only if there exists a sigma-compact open subset U of X and a phantom complex line subbundle L of epsilon over U (i.e., L is not globally trivial, but is trivial on all compact subsets of U). This phenomenon occurs whenever n >= 2 and X is a CW-complex (or a topological manifold) of dimension 3 <= d < infinity.