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The Gelfand Spectrum

In this video, we initiate Gelfand's theory of commutative C*-algebras which connects this study with the topology of compact Hausdorff spaces. This theory shows such topological spaces can be studied purely C*-algebraically and conversely, every C*-algebra can be studied through the lens of continuous functions. In this video we show that to every commutative C*-algebra A, we can naturally associate a compact Hausdorff topological space Sp(A), the Gelfand spectrum. Furthermore, there is a Gelfand transform which maps A to the C*-algebra of continuous functions on Sp(A). We look at some properties of this Gelfand transform in preparation for showing it is actually an isomorphism.