On various notions of distance between subalgebras of operator algebras

Given any irreducible inclusion of unital C^*-algebras with a finite-index conditional expectation E: , we show that the set of E-compatible intermediate C^*-subalgebras is finite, thereby generalizing a finiteness result of Ino and Watatani (from IW). A finiteness result for a certain collection of intermediate C^*-subalgebras of a non-irreducible inclusion of simple unital C^*-algebras is also obtained, which provides a C^*-version of a finiteness result of Khoshkam and Mashood (from KM). Apart from these finiteness results, comparisons between various notions of distance between subalgebras of operator algebras by Kadison-Kastler, Christensen and Mashood-Taylor are made. Further, these comparisons are used satisfactorily to provide some concrete calculations of distance between operator algebras associated to two distinct subgroups of a given discrete group.