The motivation of the study of operator spaces ties up with the notion of quantization. This young field has useful applications in operator theory and quantum physics. An inviting aspect of the theory of operator spaces is the way the classical notions in Banach space theory re-emerge as deep and beautiful ideas. For instance, the Hahn Banach theorem and the Krein Milman theorem both have a respective counterpart in quantized functional analysis. Tensor products also play a prominent role in the development of operator spaces. It gives a deep and beautiful insights of the various questions of the Banach space theory.

There are many substantial applications of the subject in different areas of Mathematics. Perhaps the most significant area in which operator spaces have been used is harmonic analysis. In particular, as the predual of the group von Neumann algebra, A(G), the Fourier algebra is an operator space in a canonical manner for every locally compact Hausdorff topological group G, which proved to be an important structure and gained a new momentum in 1995 with the work of Ruan. Using this operator space structure of A(G), Ruan added an operator space overtones to Johnsons notion of an amenable Banach algebra and introduced the concept of operator amenability. More precisely, he proved that a locally compact Hausdorff topological group G is amenable if and only if A(G) is operator amenable. Effros and Ruan went further and proved that for von Neumann algebras M and N , the predual of the spatial tensor product just the projective tensor product of the preduals M* and N* . In particular, for the locally compact Hausdorff topological groups G and H, A(G) \widehat\otimes A(H) is identified as A(G \times H).