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Tensor products of Banach spaces

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 semisimple 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 Banachalgebras. 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 normedalgebra is injective if and only if its injective tensor product with any normedalgebra is a normedalgebra. Chapter four is concerned with the question, also considered in chapter three, of whether the injective tensor product of two normedalgebras is a normedalgebra. 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.
