  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.
