Over thirty years ago, Kaplansky initiated the study of \(AW^*\)-algebras and, in particular, extended the Murray-von Neumann classification to those more general algebras. He gave a complete analysis of Type I \(AW^*\)-algebras and showed that an \({\mathcal N}\)-homogeneous Type I \(AW^*\)-algebra may be regarded as an \({\mathcal N}\times {\mathcal N}\)-matrix algebra over its centre. It follows that such an algebra is, in some sense, a tensor product of a commutative \(AW^*\)-algebra by a Type I von Neumann factor. But a general theory of tensor products for \(AW^*\)- algebras seems elusive. Apart from Kaplansky's pioneering work, the only other substantial result until recently was a theorem of Berberian. He showed that the tensor product of an arbitrary \(AW^*\)-algebra by the algebra of \(n\times n\) matrices was always an \(AW^*\)-algebra. Very recently, Hamana has made a huge advance in the tensor-product problem. Instead of considering arbitrary \(AW^*\)-algebras he concentrates most of his attention on monotone complete \(C^*\)-algebras. When A is monotone complete and M is a von Neumann algebra, Hamana constructs a tensor product \(A{\bar \otimes}M\) which is characterized by the following properties. First A\({\bar \otimes}M\) is a monotone complete \(C^*\)-algebra which is monotone generated by the algebraic tensor product \(A\otimes M\). Secondly, \(A\otimes {\mathbb{C}}\) and \({\mathbb{C}}\otimes M\) are monotone subalgebras of \(A{\bar \otimes}M.\) Thirdly if A (respectively, \(M_ 1)\) is a monotone subalgebra of the monotone complete \(C^*\)-algebra \(A_ 2\) (respectively, \(W^*\)-subalgebra of a \(W^*\)-algebra \(M_ 2)\) then \(A_ 1{\bar \otimes}M_ 1\) can be identified with the monotone subalgebra of \(A_ 2{\bar \otimes}M_ 2\) generated by \(A_ 1\otimes M_ 1.\) Hamana's tensor product is an important new tool for investigating monotone complete \(C^*\)-algebras and his papers contain a number of deep applications. His construction, which makes use of Tomiyama's notion of Fubini product, is subtle and delicate rather than transparent. Given \(A\subset L(H)\) and \(M\subset L(K),\) he constructs \(A{\bar \otimes}M\) as an operator subspace of \(L(H\otimes K)\) which is not, in general, a subalgebra of \(L(H\otimes K).\) He shows that when A is a monotone complete \(C^*\)-algebra it is possible to define a canonical product on \(A{\bar \otimes}M\) which converts it into a \(C^*\)-algebra. The main object of this note is to show that when A and M are not too large, it is possible to give a new description of \(A{\bar \otimes}M\) which, it is hoped, sheds new light on the Hamana tensor product and will be useful for applications.