If $k$ is an infinite cardinal, we say a C*-algebra $\mathcal{ A}$ is residually less than $k$ dimensional, $R_{ <k}D,$ if the family of representations of $\mathcal{ A}$ on Hilbert spaces of dimension less than $k$ separates the points of $\mathcal{ A}.$ We give characterizations of this property, and we show that if $\{ \mathcal{ A}_{ i}:i\in I\} $ is a family of $R_{ <k}D$ algebras, then the free product $\underset{ i\in I}{ \ast}\mathcal{ A}_{ i}$ is $R_{ <k}D$. If each $\mathcal{ A}_{ i}$ is unital, we give sufficient conditions, depending on the cardinal $k$, for the free product $\underset{ i\in I}{ \ast_{ \mathbb{ C}}}\mathcal{ A}_{ i}$ in the category of unital C*-algebras to be $R_{ <k}D$. We also give a new characterization of RFD, in terms of a lifting property, for separable C*-algebras.
Categories: math.OA