On various notions of distance between subalgebras of operator algebras

Given any irreducible inclusion $\text{mB}\subset \text{mA}$ of unital $C^{\ast }$-algebras with a finite-index conditional expectation $E:\text{mA}\to \text{mB}$, we show that the set of $E$-compatible intermediate $C^{\ast }$-subalgebras is finite, thereby generalizing a finiteness result of Ino and Watatani (from \cite{IW}). A finiteness result for a certain collection of intermediate $C^{\ast }$-subalgebras of a non-irreducible inclusion of simple unital $C^{\ast }$-algebras is also obtained, which provides a $C^{\ast }$-version of a finiteness result of Khoshkam and Mashood (from \cite{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.