1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Appendix B

Operator Spaces,

Completely Bounded

Maps and Duality

Here is a crash course on the operator space theory needed in these notes.

Completely bounded maps and perturbation results.

Definition B.1. An operator space Xis a closed subspace of a C*-algebra
A. Note that for each n EN, Mn(X) inherits a norm from Mn(A). Let r.p be
a linear map from an operator space X c A into an operator space Y c B.
We say r.p is completely bounded (abbreviated c.b.) if

llr.pllcb :=sup 111.fJn: Mn(X) -t Mn(Y) II < oo,

nEN

where 1.fJn is defined by 1.fJn([xi,j]) = [r.p(xi,j)] for x = [xi,j] E Mn(X). The

number llr.pllcb is called the cb norm of r.p. We say r.p is completely contractive

(c.c.) if llr.pllcb::::; 1; r.p is completely isometric (c.i.) if 1.fJn: Mn(X) -t Mn(Y)

is isometric for every n. We denote by CB(X, Y) the Banach space of all

c.b. maps from X into Y, equipped with the cb norm.

Example B.2. A bounded linear functional r.p on an operator space X c A

is c.b. with llr.pllcb = 111.fJll· Indeed, for any [xi,j] E Mn(X), we have

111.fJn([xi,jDllMn(C) = sup{l(r.pn([xi,j])77,e)I: ~,77 E c;, llell = 1=117711}

= sup{lr.p(L.::ei7]jXi,j)I: L 1eil^2 = 1 = L l77jl^2 }

i,j i j

• 449