#### Title

Examples of Banach spaces that are not Banach algebras

#### Date of Completion

January 2007

#### Keywords

Mathematics

#### Degree

Ph.D.

#### Abstract

Let *A ^{p}* be the Banach space of all continuous functions on the torus whose Fourier coefficients are in ℓ

*. We show that*

^{ p}*A*is not an algebra for all 1 <

^{p}*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*and whether or not it is an algebra. Of particular interest is the space

^{p}*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. ^

#### Recommended Citation

Mullen, Ryan, "Examples of Banach spaces that are not Banach algebras" (2007). *Doctoral Dissertations*. AAI3265786.

http://digitalcommons.uconn.edu/dissertations/AAI3265786