Abstract
Chapter one consists of a general discussion of
tensor products.
Chapter two is concerned with the relationship between
tensor products and the approximation property. In
Theorem 2.1 we give an equivalent condition to the
approximation property which is due to Grothendieck.
In Theorem 2.5 we prove that every complex Banach space
is isometrically isomorphic to a complemented subspace
of a uniform algebra. From this, we prove in Theorem 2.6
that there exists a uniform algebra not having the
approximation property. Tomiyama has shown that if A and B
are semi-simple commutative Banach algebras, and either
A or B has the approximation property, then A ⊗^ B is
semi -simple. In Theorem 2.8 we establish a converse to
this result, namely that if A is a commutative Banach algebra
not having the approximation property, then there is a uniform
algebra B such that A ⊗^0 B is not semi -simple. We next
discuss the c- product and the slice product, and their
relationships with the injective tensor product and with
the approximation property. Then, in Theorem 2.11,
we prove that a uniform algebra A has the approximation
property if and only if A ⊗^ B = A # B for all uniform
algebras B.
In chapter three we consider injective algebras.
Using techniques similar to those used in the proof of
Theorem 2.5, we give a proof in Theorem 3.2 of
Varopoulos's characterisation of injective commutative
Banach-algebras. This states that a commutative Banachalgebra
A is injective if and only if there exists a
uniform algebra B, a bounded algebra homomorphism h of
B onto A, and a bounded linear operator j of A into B
such that hₒj = Iₐ. In Theorem 3.4 we prove a
sharpening of Varopoulos's result that a normed-algebra
is injective if and only if its injective tensor product
with any normed-algebra is a normed-algebra.
Chapter four is concerned with the question,
also considered in chapter three, of whether the injective
tensor product of two normed-algebras is a normed-algebra.
We show that this is the case for the tensor product
1ₚ ⊗ᵛ lq (where p or q ≤ 2), and for the tensor product
of two Banach- algebras which are ℓ₁ spaces.
In chapter five we consider measures orthogonal
to injective tensor products of uniform algebras, and
we obtain an analogue of Cole's decomposition theorem
for orthogonal measures to the bidisc algebra.
Through a general study of bands, we set up the
decomposition in Lemma 5.4, and prove that this
decomposition is of the form we want in Theorem 5.7.
This then gives us our main result in Theorem 5.8.
