A Complete Characterization of Cartan Inclusions of Finite Dimensional $C^{\ast }$-algebras

We give a complete characterization of Cartan inclusions of finite dimensional $C^{\ast }$-algebras in terms of their inclusion matrices. More precisely, for a unital inclusion $\mathcal{B}\subseteq \mathcal{A}$ with inclusion matrix $\mathrm{\Lambda }=({\mathrm{\Lambda }}_{ij})$, where Cartan means that $\mathcal{B}$ is a \emph{generalised Cartan subalgebra} of $\mathcal{A}$ in the sense of Exel, we prove that the inclusion is Cartan if and only if $$\sum _i{\mathrm{\Lambda }}_{ij}\le 1$$ for every $j$. We call matrices satisfying this condition \emph{multiplicity free}. Thus, our characterization provides a purely combinatorial criterion for determining when a finite dimensional inclusion is Cartan. We further prove that every Cartan inclusion admits a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$. Conversely, we show that, for unital inclusions of finite dimensional $C^{\ast }$-algebras, the uniqueness of the conditional expectation is sufficient for the inclusion to be Cartan. Consequently, a unital inclusion $\mathcal{B}\subseteq \mathcal{A}$ of finite dimensional $C^{\ast }$-algebras is Cartan if and only if there exists a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$.