TL;DR
This paper investigates bases for finite index inclusions of $II_1$ factors and finite dimensional $C^*$-algebras, exploring their properties, automorphisms, and characterizations of basic constructions.
Contribution
It introduces bases that behave well under basic construction towers and characterizes automorphisms and multistep basic constructions using these bases.
Findings
Bases behave nicely with respect to basic construction towers
Characterization of automorphisms compatible with Jones' tower
Description of multistep basic constructions using bases
Abstract
In this paper we examine bases for finite index inclusion of factors and connected inclusion of finite dimensional - algebras. These bases behave nicely with respect to basic construction towers. As applications we have studied automorphisms of the hyperfinite factor which are `compatible with respect to the Jones' tower of finite dimensional -algebras'. As a further application, in both cases we obtain a characterization, in terms of bases, of basic constructions. Finally we use these bases to describe the phenomenon of multistep basic constructions (in both the cases).
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