This fascinating paper is concerned with approximation properties for operator spaces. The paper describes five approximation constants \(\Lambda(X)\), \(\Lambda_1(X)\), \(\Lambda_2(X)\), \(\Lambda_3(X)\) and \(\Lambda_4(X)\) associated with each operator space \(X\). The first coincides with the Haagerup invariant of a \(W^*\)-algebra \(M\) when \(X\) is the predual of \(M\), \(\Lambda_3(X)\) is new and the rest have been considered by other authors. The first striking result is that, for an arbitrary operator space \(X\), all the constants coincide. This is then applied to the operator space projective tensor product \(X\widehat\otimes Y\) of two operator spaces \(X\) and \(Y\). The key result is that \(\Lambda(X)\Lambda(Y)\geq \Lambda(X\widehat {\otimes} Y)\) and an important corollary shows that, for \(W^*\)-algebras \(M\) and \(N\) the Haagerup invariant of their spatial tensor product \(M\overline {\otimes} N\) is the product of those for \(M\) and \(N\).