On the C-Projectivity of ideals in Banach algebras

Abstract

The notion of projective Banach module was defined by Helemskii in [1]-the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.For a given constant C > 0 the Banach module Zover a Banach algebra A, (abbreviated below as "module"), is called C-projective gives us the more useful equivalent definition of C-projectivity. Namely, a module X is C-projective if and only if the morphism of external multiplication n : A ® X ->• X, defined by the formula n(a <g> x) = ax, has a right inverse morphism p such that ||p|| < C. Here the symbol® denotes the projective tensor product of Banach spaces [4]. If ||p|| = C and there is no right inverse with a norm smaller than C, then it is natural to say that X is exactly C-projective. In this paper we give answers to two questions that (directly or not) were put in [3]. First, for arbitrary C > 1, we give an example of an exactly C-projective Banach ,4-module. (Moreover, it is a maximal ideal in a uniform algebra A.) Note that C-projectivity is impossible for C < 1 and for C = 1 there exist trivial examples: consider for example any maximal ideal in the disc-algebra, corresponding to an inner point of the disc. Second, we shall show that C-projectivity does not possess the same "continuity property" as C-flatness [3]: that is, there exists a module (again a maximal ideal in the uniform algebra) that is (C + c)-projective for all e > 0 but not C-projective.As usual, we denote by A(E) the uniform algebra of functions that are continuous on the given compact subset £ c C and analytic in its interior. EXAMPLE 1. Consider the compact subset A: = D U £ o / C x R , where D -{(z, 0): \z\ < 1} is the closed disc and E = {(z, i) : |C|~' < \z\ < 1, 0 < t < 1} is the cylindric annulus. (We denote by E, its section, where / is constant.) Consider the uniform algebra Proof. (1) Consider two functions: h e M such that h(z, t) = z for (z, t) € K and f{z, t) = l/z for (z, t)eK\O.Note that \\h\\ = 1 and, for each meM,fm is defined on K\O and we extend the definition to K by continuity. We have ||/m|| < C||/w||, because | f\ < C on f This work was partially supported by ISF (Soros foundation) grant M95300 and Russian RFFI grant 93-01-00156.