Compactness and approximation property with respect to an operator space Khye Loong Yew 46B2846L0746M0547L25tensor productsapproximation propertiesoperator spacescompletely bounded maps For any operator space $Z$, the notions of $Z$-compactness and $Z$-approximation property are introduced in the category of operator spaces. Several interesting phenomena when $Z$ is the full group $C^*$-algebra of the free group on countably infinitely many generators, compact operators and bounded linear operators on the Hilbert space $\ell_2$ are presented here. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3149 10.1512/iumj.2007.56.3149 en Indiana Univ. Math. J. 56 (2007) 3075 - 3128 state-of-the-art mathematics http://iumj.org/access/