BANACH ALGEBRAS, C*-ALGEBRAS AND THE GNS CONSTRUCTION

Abstract

The first part of the project where we talk about Banach algebras basically comprises of definition of Banach algebra, discussions on commutative, non-commutative, unital, non-unital Banach algebras and unitilization of a non-unital Banach algebra. It also comprises of discussions on invertibility and non-invertibility of elements. In this context, we prove an important result that "the set of invertible elements G(A ) in a unital Banach algebra A is open in A ". We also define spectrum of an element, resolvent set and spectral radius of an element. This paves the way for an important proposition that "For any x in an unital Banach algebra A , the spectrum of element x is a non-empty compact subset of C with σA (x) ⊆ {λ ∈ C: |λ| ⩽ ∥x∥} where σA (x) denotes the spectrum of x". In the subsection comprising of Gelfand theory, we discuss about the interplay between the maximal ideals of a unital Banach algebra, the multiplicative linear functionals and associated spaces. The pivotal constituent of Gelfand Theory is the celebrated Gelfand Mazur theorem which states that "Let A be a unital Banach algebra where each non-zero element is invertible.Then A ≃ C". The second part of the project comprises of discussions on C*-algebras. C∗-algebras is a particular type of Banach algebra and it has intimate connections with the theory of operators on Hilbert space. If H is a Hilbert space, then B(H ) is an example of C*-algebra where B(H ) is the set of all bounded linear operators onH . In this section, we have discussions about commutative, non-commutative, unital and non-unital C∗-algebras. The most important theorems of this section is the Gelfand Naimark Theorem for a commutative unital C∗-algebra which states that "Let A be a commutative unital C∗-algebra. Then the Gelfand transform is a ∗-isometric isomorphism from A onto C(Sp(A ))" (where Sp(A ) denotes the spectrum of A ,consisting of non-zero complex valued homomorphisms and C(Sp(A )) is the space of all continuous functions on Sp(A ) ). This section also includes definitions, examples of C∗-algebra, homomorphisms and states on a C∗-algebra. The last part of the project which is the main part of the project is of utmost importance. With short discussions on representations of a unital C∗-algebra and cyclic vector, we finally reach the crux of the section where the Gelfand, Naimark, Segal Construction (the GNS Construction) plays the role of the central ingredient. The GNS Construction depicts the construction of a representation of a C∗-algebra A from a state on A .This will induce the inference that any C∗-algebra can be perceived as a C∗-algebra of operators on a Hilbert space.