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        The Gelfand-Naimark theorem for commutative Banach star algebras

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        Bachelor Thesis Gelfand-Naimark Robin Verstraten.pdf (459.8Kb)
        Publication date
        2017
        Author
        Verstraten, R.C.
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        Summary
        We start by studying properties of several kinds of algebras, taking a look at the spectrum, ideals and abelian algebras. Then we prove the Gelfand-Naimark Theorem for commutative Banach star algebras (also known as abelian C*-algebras). After that, we prove some basic theorems about Riemann integration of Banach valued functions. We then study some applications of the Gelfand-Naimark Theorem where we start by studying the functional calculus, in particular the Riesz functional calculus and its extension to C*-algebras. We then take a look at positive elements, representations of C*-algebras and in particular the Gelfand-Naimark-Segal construction. Lastly, we study spectral measures and, using representations, we prove the spectral theorem for bounded normal operators on a Hilbert space.
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        https://studenttheses.uu.nl/handle/20.500.12932/26989
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