Banach and C*-algebras -- Christopher Lopez
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Banach and C*-algebras arise in functional analysis as one studies certain 
function spaces defined over a topological space. The spectra of these functions generalize the sets of eigenvalues of linear transformations on a vector space. We will study the relationship between the maximal ideals and spectra of commutative unital Banach and C*-algebras. We will be able to show that a compact Hausdorff space X is homeomorphic to the maximal ideal space of the Banach algebra of continuous functions on X. Along the way, we will encounter results of functional analysis such as the Hahn-Banach and Banach-Alaoglu theorems.