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Title: Tensor products of Banach spaces
Author: Milne, Hamish K.
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1974
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available