Banach algebras

The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. This particular paper focuses on Gelfand theory — the relation between multiplicative linear functionals on a commutative Banach algebra and its maximal ideals, as well as with the spectra of its elements. Most of the content of chapters 1 thorough 3 is meant, in one way or another, to lead towards this theory. The central ingredient of Gelfand theory is the well-known Gelfand-Mazur theorem which says that if a Banach algebra is...Show moreThe study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. This particular paper focuses on Gelfand theory — the relation between multiplicative linear functionals on a commutative Banach algebra and its maximal ideals, as well as with the spectra of its elements. Most of the content of chapters 1 thorough 3 is meant, in one way or another, to lead towards this theory. The central ingredient of Gelfand theory is the well-known Gelfand-Mazur theorem which says that if a Banach algebra is a division algebra then it is isomorphic to C. The first chapter is a purely algebraic one and provides us with all the necessary algebraic techniques, particularly concerning algebras without identity. The second and third chapters introduce normed algebras and Banach algebra and other concepts like the spectrum, and prove several important results among which the Gelfand-Mazur theorem. The fourth chapter is the pivotal one — where Gelfand theory is developed. In the fifth chapter several examples of Banach algebras are discussed in detail, together with their Gelfand representations. Some practical applications of the theory are also mentioned, among which Wiener’s famous theorem about zeroes of functions with absolutely Fourier series, proven entirely from the context of Banach algebras.Show less

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Banach algebras