The Product/Sum Identity for Logarithmic Residues in Higher Dimensions

In the scalar case, the logarithmic residue of the product fg of two analytic functions f and g is the sum of the logarithmic residues of f and g . Here attention is paid to what happens in the higher dimensional situation where f and g have values in a complex Banach algebra. In the context of giving some background information on vector-valued logarithmic residues, a couple of open problems are indicated.