1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.8. Nuclearity and tensor products 101

is continuous with respect to the spatial tensor product norm and let 7r: M -t

lffi(JC) be any normal representation. Then the product map

(7r o <p) x l7r(M)': A 0 7r(M)' -t lffi(JC)

is also min-continuous.^24

Proof. Any normal representation of M can be identified with the cut-down

by a projection in the commutant of the representation M®lJC c lffi(H®JC).

Hence it suffices to show that the product map with the commutant in this

particular representation is min-continuous.

Since (M ® lJC)' n lffi(H ®JC) = M' ® lffi(JC) - just think of lffi(H ®JC) as

matrices with entries in lffi(H) - we thus have to show that

(<p® llB(JC)) x lM'®IBl(JC): A0 (M' ®lffi(JC))-tlffi(H®JC)
is min-continuous. But, except for the horrific notation required, this is
easy since (<p ® llB(JC)) x lM'®IBl(JC) is a point-strong limit of min-continuous
maps (with uniformly bounded norms). More precisely, if P E lffi(JC) is a
finite-rank projection, then the map
(<p ® llB(PJC)) x lM'@IB(PJC): A 0 (M' ® lffi(PJC)) -t lffi(H ® PJC)
is min-continuous and its norm is bounded by ll'P x lM' II because it can be
identified with
(<p x lM') ® idlB(PJC): (A 0 M') 0 lffi(PJC) -t lffi(H ® PJC)
and Exercise 3.5.1 then comes into play. Finally, taking a net {P.A} of finite-
rank projections which converge to lJC in the strong operator topology and
fixing

x =Lai ®7i E A0 (M' ®lffi(JC)),

it is easy to check that

( <p® llB\(P,\JC)) X lM'@lB\(P>.!C) ( (17-l®P.A)x(l?-l®P.A)) -t ( <p® llB\(JC)) X lM'®IBl(JC) ( x)

in the strong operator topology. This completes the proof. D

We are now ready for an important theorem of Kirchberg. Though the

proof is not so long, it is delicate and deliciously technical. Bon appetit!

Theorem 3.8.5. Let <p: A -t MC lffi(H) be a u.c.p. map from a unital

C*-algebra A to a von Neumann algebra M. Then <p is weakly nuclear if

and only if the product map <p x lM': A 0 M' -t lffi(H) is continuous with

respect to the spatial tensor product norm.

24That is, "continuous with respect to the minimal norm."