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Let A ^p be the Banach space of all continuous functions on the torus whose Fourier coefficients are in [cursive l] ^p . We show that A ^p is not an algebra for all 1 < p < p ₀ , for a certain p ₀ , 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 A ^{p ,∞} : how it relates to A ^p and whether or not it is an algebra. Of particular interest is the space A ¹ _,∞ which we show is not an algebra, which is a curiosity given that A ¹ is a well known algebra. We also give examples to show that all of these spaces are indeed different.