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Character space of generated normal algebra is operator spectrum

Statement

Assume AC. For a bounded normal operator $T$ on a nonzero complex Hilbert space, the map $χ \mapsto χ (T)$ is a homeomorphism from the character space $Δ (C^{*} (I, T))$ onto the operator spectrum $σ (T)$ .

Facts & Assumptions

[A1]

For normal $T$ the algebra $C^{*} (I, T)$ is a nonzero unital commutative C*-algebra contained in $B (H)$ with the same identity, and its elements are norm limits of $*$ -polynomials in $T$ (C star algebra generated by a normal operator).

[A2]

The character space $Δ (A)$ of a commutative unital complex Banach algebra carries the pointwise-evaluation topology, in which each evaluation $χ \mapsto χ (a)$ is continuous; for a nonzero commutative unital C*-algebra it is nonempty and compact Hausdorff (Character and maximal ideal space, Maximal ideal space is compact Hausdorff).

[A3]

$σ_{C^{*} (I, T)} (T) = {χ (T) : χ \in Δ (C^{*} (I, T))}$ (Spectrum as character values).

[A4]

$σ_{C^{*} (I, T)} (T) = σ_{B (H)} (T) = σ (T)$ by spectral permanence and the bounded inverse theorem (Spectral permanence for unital c star subalgebras, Spectrum and resolvent of a bounded operator, Bounded inverse theorem, Bounded Hilbert operators form a C star algebra).

[A5]

Characters satisfy $χ (a^{*}) = \overline{χ (a)}$ , and a character is continuous for the norm (Characters on a unital commutative C star algebra preserve star, Maximal ideal space is compact Hausdorff).

[A6]

The spectrum is a subset of the metric space $C$ with its usual subspace topology (Spectrum and resolvent of a bounded operator, The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane). It is Hausdorff: distinct $z, w$ have disjoint relative open balls of radius $∣ z - w ∣ / 3$ , by the triangle inequality.

[A7]

The continuous image of an arbitrary compact space is compact, and every compact subset of a Hausdorff space is closed; hence a continuous bijection from a compact space onto a Hausdorff space is a homeomorphism (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones).

[A8]

AC is the global hypothesis (The Axiom of Choice).

Proof

technique · direct

Given: A nonzero complex Hilbert space $H$ and a bounded normal operator $T \in B (H)$ , with $Δ : = Δ (C^{*} (I, T))$ and $Φ (χ) : = χ (T)$ .

1.1

$Φ$ maps $Δ$ onto $σ (T)$ : the character values of $T$ in $C^{*} (I, T)$ are exactly that spectrum, which equals the operator spectrum by spectral permanence.

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1.2

$Φ$ is continuous: evaluation at $T$ is continuous in the pointwise-evaluation topology.

1.3

$Φ$ is injective: if $χ (T) = ψ (T)$ then also $χ (T^{*}) = \overline{χ (T)} = \overline{ψ (T)} = ψ (T^{*})$ , so the two continuous characters agree on $I$ , $T$ and $T^{*}$ and hence, by continuity and multiplicativity, on the norm closure of the unital $*$ -algebra they generate, which is $C^{*} (I, T)$ .

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2.1

The source $Δ$ is compact Hausdorff and the target $σ (T)$ is Hausdorff, so the continuous bijection $Φ$ carries closed subsets of $Δ$ to compact, hence closed, subsets of $σ (T)$ ; therefore $Φ^{- 1}$ is continuous and $Φ$ is a homeomorphism.

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Sources

Dana P. Williams, Lecture Notes on the Spectral Theorem, Example 4.9, pp.13–15 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.69 and Theorem 5.70, printed pp.268–273 (standard reference, not scraped)