Reformulation of the extension of the -metric for

The classical ν -metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in Ball (2012) [1]. In Sasane (2012) [11], an extension of the classical ν -metric was given when the underlying ring of stable transfer functions is the Hardy algebra, H ∞ . However, this particular extension to H ∞ did not directly fit in the abstract framework given in Ball (2012) [1]. In this paper we show that the case of H ∞ also fits into the general abstract framework in Ball (2012) [1] and that the ν -metric defined in this setting is identical to the extension of the ν -metric defined in Sasane (2012) [11]. This is done by introducing a particular Banach algebra, which is the inductive limit of certain C ∗ -algebras.