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

Abstract:

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.