Reformulation of the extension of the ν-metric for H∞

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