Examples of Banach spaces that are not Banach algebras

Date of Completion

January 2007






Let Ap be the Banach space of all continuous functions on the torus whose Fourier coefficients are in ℓ p. We show that Ap is not an algebra for all 1 < p < p 0, for a certain p0, 1 < p 0 < 2. This is done through a series of attempts which might suggest that the example used is the best one possible. One of the attempts is using the Rudin-Shapiro polynomials and as an aside some new properties of these polynomials are given. We also discuss the space Ap ,∞: how it relates to Ap and whether or not it is an algebra. Of particular interest is the space A1,∞ which we show is not an algebra, which is a curiosity given that A1 is a well known algebra. We also give examples to show that all of these spaces are indeed different. ^