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By Robert C. Gunning, Hugo Rossi

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If x ∈ A, x sup = x if and only if x2 = x 2 for all k ≥ 1. b. ΓA is an isometry if and only if x2 = x 2 for all x ∈ A. © 2016 by Taylor & Francis Group, LLC Banach Algebras and Spectral Theory Proof. If x sup = x then k x2 k 11 ≤ x 2k = x 2k sup k k = x2 k sup k ≤ x2 , k so x2 = x 2 . 13(d). This proves (a), and k k (b) follows since if x2 = x 2 for all x then x2 = x 2 for all x and k (by induction on k). We now come to the most fundamental result of Gelfand theory. 20 Theorem (The Gelfand-Naimark Theorem).

2016 by Taylor & Francis Group, LLC Banach Algebras and Spectral Theory 17 The finite-dimensional spectral theorem, in its simplest form, says that if T is a self-adjoint operator on a finite-dimensional Hilbert space H, there is an orthonormal basis for H consisting of eigenvectors for T . In this form the theorem is false in infinite dimensions, where self-adjoint operators need not have any eigenvectors at all. ) However, there are ways of reformulating the theorem that do generalize. Formulation I.

Let A be a commutative C* subalgebra of L(H) containing I. There is a semi-finite measure space (Ω, M, µ), a unitary map U : H → L2 (µ), and an isometric ∗homomorphism T → φT from A into L∞ (µ) such that U T U −1ψ = φT ψ for all ψ ∈ L2 (µ) and T ∈ A. Ω can be taken as the disjoint union of copies of the spectrum Σ of A in such a way that µ is finite on each copy and φT = T on each copy. Proof. 33). Then, for any T ∈ A, Tv 2 = T ∗ T v, v = |T |2 dµ. , so T v → T is a welldefined linear isometry from Av into L2 (µ), and it extends uniquely to a linear isometry U : H → L2 (µ).

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