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

Abstract

For a C*-algebra A we consider the problem of when the set $TM_0(A)$ of all two-sided multiplications $x \mapsto axb$ ($a,b \in A$) on A is norm closed, as a subset of B(A). We first show that $TM_0(A)$ is norm closed for all prime C*-algebras A. On the other hand, if $A\cong \Gamma_0(E )$ is an n-homogeneous C*-algebra, where E is the canonical $\mathbb{M}_n $-bundle over the primitive spectrum X of A, we show that $TM_0(A)$ 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 E over U (i.e. L is not globally trivial, but is trivial on all compact subsets of U). This phenomenon occurs whenever $n \geq 2$ and X is a CW-complex (or a topological manifold) of dimension $3 \leq d<\infty$.