This paper is an investigation of right modules over a
B
∗
{B^\ast }
-algebra B which posses a B-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a
W
∗
{W^\ast }
-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and
W
∗
{W^\ast }
-algebras. The extension of an inner product module over B by a
B
∗
{B^\ast }
-algebra A containing B as a
∗
^\ast
-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.
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