Continuing with the linear algebra theme, let’s consider the space of all matrices with real entries. This is a 4-dimensional vector space; we can treat a matrix as a vector . So a general linear map can be described by a matrix. That is, the space of linear maps is 16-dimensional.
However, some linear maps arise more naturally than others; they are somehow more “matricial”. For example, left multiplication by a fixed matrix . Or right multiplication by a fixed . Or, to hit both with one stone, the two-sided multiplication map defined by . The matrix of this linear map is the Kronecker product :
Observe that the set of Kronecker products is not closed under addition. This is somehow unsatisfactory. It would be nice if “matricial” transformations, whatever they are, formed a linear subspace of . This can be fixed by generalizing the two-sided multiplication map: any representation of on some vector space and any pair of operators , , give rise to a linear map . Given another triple , we can take the direct product of representations and arrange the operators accordingly:
Hurray, we have a linear subspace of ! Now, what is this subspace? Maybe all ? Hm.
Let be the space of square matrices of size . We can think of them as block matrices with blocks of size . A linear map induces : just apply to every block. If comes from a triple as above, then can be obtained if we let act on blockwise and multiply on left and right by and . In this case the norm of is bounded by — the bound is independent of . Note that the norm of is taken with respect to the matrix norm on
An operator such that is called completely bounded. We just saw that any operator of the form is completely bounded. The converse is also true; it was proved independently by Haagerup, Paulsen and Wittstock (and not just for matrices, of course).
This certainly helps. Still, which operators on are completely bounded? So far we saw that they form a linear space of dimension between 8 and 16…