Title: Gelfand Duality

Abstract: If X and Y are locally compact Hausdorff spaces, and C_0(X) and C_0(Y) are the rings of complex-valued functions over the respective spaces that vanish at infinity, then X and Y are homeomorphic if and only if C_0(X) and C_0(Y) are isomorphic as rings. A question arises: is a commutative ring satisfying similar axioms to a ring of complex-valued functions vanishing at infinity (even with no reference to an underlying space in its definition) canonically isomorphic to C_0(X) some locally compact Hausdorff space X? As it turns out, yes! In this talk, I will introduce and outline the proof of Gelfand duality, a foundational correspondence between commutative C*-algebras and locally compact Hausdorff spaces. I will then exhibit the character space — the key ingredient to the correspondence — for some standard commutative C*-algebras. In the process, we will obtain a dictionary between properties and constructions of locally compact Hausdorff spaces and those of commutative C*-algebras. No prior knowledge of operator algebras will be assumed for this talk.

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