P. G. Dixon

Left approximate identities in algebras of compact operators on Banach spaces

We study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras ( X )of all compact operators on a Banach space X and ℱ( X ) − of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in ( X )(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

Topologically Irreducible Representations and Radicals in Banach Algebras

It is shown that the topologically irreducible representations of a normed algebra define a certain topological radical in the same way that the strictly irreducible representations define the Jacobson radical and that this radical can be strictly smaller than the Jacobson radical. An abstract theory of ‘topological radicals’ in topological algebras is developed and used to relate this radical to the Baer radical (prime radical). The relations with topologically transitive representations and standard representations in the sense of Meyer are also explored. 1991 Mathematics Subject Classification: 46H15, 46H25, 16Nxx.

Non-closed sums of closed ideals in Banach algebras

A major diculty in Banach algebra theory is that sums of closed ideals need not be closed. We survey the known results and present examples showing that they are in most directions the best possible. We also give a new sucient condition for closure in the uniform algebra setting.

Factorization and unbounded approximate identities in Banach algebras

Cohens Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A . Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

Graded Banach algebras associated with varieties of Banach algebras

The concept of ‘varieties of Banach algebras’ was introduced by the author (Quart. J. Math. Oxford (2), 27 (1976), 481–487). Here we develop the theory, producing analogues of the ‘relatively free’ objects in varieties of universal algebras. We take as a test question the problem of describing all varieties which are closed under bicontinuous isomorphisms (i.e. varieties which are also semivarieties), and show that this may be answered easily with the aid of these new concepts. (The answer is, as expected, just those varieties which are ‘algebraically defined’.)

Unitary dilations, polynomial identities and the von Neumann inequality

Let p(X 1 , …, X n ) be a polynomial in the commuting indeterminates X 1 , …, X n . Define

A lower bound for the L 1 -norm of certain exponential sums

Littlewood [5, Problem 4.19, originally 4] conjectured that there is an absolute constant C > 0 such that, for every sequence of distinct integers n 1 , n 2 , n 3 , …, if then Cohen [2] showed for some absolute constant C , with b = 1/8. Davenport [3] gave a more explicit version of Cohens proof and improved the estimate to b = 1/4. Pichorides [6] added another refinement to obtain b = ½, and has, more recently, obtained ‖ f N ‖ 1 > C (log N ) 1/2 . This seems to be the best estimate so far without restriction on the sequences. We shall show that the methods of Davenport and Pichorides may be extended to obtain better results for certain classes of sequences. Specifically, we prove the following theorems.