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For a   $C^*$  -algebra   $A,$   we consider the problem when the set   $\mathop{\mathrm{TM}}\nolimits_0(A)$   of all two-sided multiplications   $x \mapsto axb$    $(a,b \in A)$   on   $A$   is norm closed, as a subset of   $\mathcal{B}(A)$  . We first show that   $\mathop{\mathrm{TM}}\nolimits_0(A)$   is norm closed for all prime   $C^*$  -algebras   $A$  . On the other hand, if   $A\cong \Gamma_0(\mathcal{E} )$   is an   $n$  -homogeneous   $C^*$  -algebra, where   $\mathcal{E} $   is the canonical   $\mathbb{M}_n $  -bundle over the primitive spectrum   $X$   of   $A$  , we show that   $\mathop{\mathrm{TM}}\nolimits_0(A)$   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   $\mathcal{L} $   of   $\mathcal{E} $   over   $U$   (i.e.,   $\mathcal{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   $3 \leq d&lt;\infty$  .

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