Chapter 3
Tensor Products
This chapter is devoted to tensor products in the C -context. Those with a
low tolerance for operator algebras often find it unfortunate that the theory
is subtle, tricky and requires great care. We prefer to embrace this fact.
Indeed, if tensor products behaved too nicely, then operator algebras would
be far less interesting. That would be unfortunate.
Unlike the previous chapter, this one contains several major theorems.
Of course, there are also lots of preliminaries, remarks, exercises and dis-
cussions of various land mines that one should avoid. Sections 3.1, 3.2 and
3.3 lay out the basics of tensor products, starting in the algebraic realm
and moving to the analytic world. For the experts these three sections
just establish notation. In Section 3.4 comes the first important theorem:
the spatial norm is the smallest possible C-norm on the tensor product
of two C -algebras. Section 3.5 contains some important continuity results
for maps on tensor products; we use these facts over and over. In Sections
3.6 and 3.7 we investigate two subtleties which make C-tensor theory quite
different from the algebraic variety. Though these sections are mostly about
examples and counterexamples, there is one important fact that deserves
mention: The Trick described in Proposition 3.6.5 is extremely useful and
will be used many times. Perhaps the two most important theorems come
in Sections 3.8 and 3.9 where we give tensor product characterizations of
the classes of nuclear and, respectively, exact C* -algebras.
3.1. Algebraic tensor products
For those who haven't studied algebraic tensor products, we now describe
how they are constructed and some of their basic properties. There are three
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