66
Mn(C) 8 A~ Mn(A) given by
L ei,j 0 ai,j 1---+ [ ai,j].
i,j- Tensor Products
Exercise 3.JL.4. For an arbitrary nonunital algebra C let C denote the
unitization. Assume A and B are both nonunital and discuss the relations
between A 8 B, (.i8B), A 8 B and A 8 B. For example, which ones can
be identified with subalgebras of the others? How about ideals? What are
the corresponding quotients?
Exercise 3.1.5. Let A and B be C-algebras. Prove that A0B* separates
points of A 8 B -i.e., show that for every 0 =f. x E A 8 B there exist
linear functionals <p1, ... , 1Pn on A and '1/J1, ... , 'I/Jn on B such that
n
(I: 1Pi 0 'I/Ji) (x) =1-o.
i=l
In fact, the elementary tensor product functionals { <p 8 'ljJ : <p EA, 'ljJ EB}
separate points.
Though it is technical and not particularly interesting, we will need the
following exercise later on (hence the long hint).
Exercise 3.1.6. Assume A is nonunital, ?TA: A ----+ lIB(H) and ?TB: B ----+
JIB(H) are *-homomorphisms with commuting ranges and the product homo-
morphism ?TA x ?TB: A 8 B ----+ JIB(H) is injective. Prove that the product
homomorphism iiA x ?TB: AGE----+ JIB(H) is also injective. (Hint: If x E AGE
i~ an element such that iiA x 1TB(x) = 0, then - since A 8 Bis an ideal in
A 8 B - for every y EA 8 B we have that xy = 0. Now write
n
x = L:ai 0 bi
i=l
where the bi's are linearly independent. Let { ek} c A and {fk} c B be
approximate units and we get that
n
0 = x(ek 0 fk) = L(aiek) 0 (bdk)
i=l
for all k. But for large k, {bilk, ... , bnfk} must also be a linearly independent
set.)
3.2. Analytic preliminaries
Our first goal is to make the tensor product of two Hilbert spaces 1i8K into a
Hilbert space. While 1i8K does have a natural positive definite sesquilinear