Boundary representations on $C\sp{\ast}$-algebras with matrix units

Let $\mathcal{A}$ be a ${C^\ast}$-algebra with unit, let $\mathcal{S}$ be a linear subspace of $\mathcal{A} \otimes {M_n}$ which contains the natural set of matrix units and which generates $\mathcal{A}$ as a ${C^\ast}$-algebra. Let $\mathcal{J}$ be the subset of $\mathcal{A}$ consisting of entries of matrices in $\mathcal{S}$. Then the boundary representations of $\mathcal{A} \otimes {M_n}$ relative to $\mathcal{S}$ are parametrized by the boundary representations of $\mathcal{A}$ relative to $\mathcal{J}$. Also, a nontrivial example is given of a subalgebra of a ${C^\ast}$-algebra which possesses exactly one boundary representation.