What Is an Operator Space?

These are notes for a lecture delivered on 12 May, 2008, in a graduate course on operator algebras in Berkeley. The intent was to give a brief introduction to the basic ideas of operator space theory. The notes were hastily written and have not been carefully checked for accuracy or political correctness. 1. An overview of operator spaces We learn early on that every Banach space S is isometrically isomorphic to a linear subspace of a unital commutative C∗-algebra, namely C(X) for some compact Hausdorff space X. For example, one can take X to be the unit ball of the dual of S, endowed with its weak∗-topology. An operator space is a complex-linear space S ⊆ B(H) of operators on some Hilbert space H that is closed in the norm of B(H). Operator spaces are the objects of a category, but we have not yet defined the maps of this category. In particular, we have not said precisely when two operator spaces S1 and S2 are considered equivalent. For example, one might consider S1 ⊆ B(H1) and S2 ⊆ B(H2) to be equivalent if they are isometrically isomorphic as Banach spaces. This is the notion of equivalence that results from declaring the maps of hom(S1,S2) to be linear mappings L : S1 → S2 such that ‖L(A)‖ ≤ ‖A‖ for all A ∈ S1. But that category is only a disguised form of the more familiar category of Banach spaces (with contractions as maps) and we don’t need subspaces of B(H) to describe Banach spaces as the first paragraph shows. In fact, once we have made a proper definition of the maps of the category of operator spaces, we will find that operator spaces are a subtle refinement of Banach spaces, whose objects S carry along with them nonclassical features that are connected with noncommutativity of operator multiplication. Some analysts like to think of operator space theory as “quantized functional analysis” in the sense that the resulting category is a noncommutative refinement of the classical category of Banach spaces. 2. Completely contractive maps We now turn to the issue of properly defining the maps of the category of operator spaces so as to cause these remarks to have some concrete meaning.