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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 $Φ (χ) : = χ (T)$ .

1.1A1A3A4

$Φ$ 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.

1.2A2

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

1.3A1A5algebra

$Φ$ 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 $C^{*} (I, T)$ .

2.1step 1.1step 1.2step 1.3A2A6A7A8∎

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.

Character space of generated normal algebra is operator spectrum

Statement

Facts & Assumptions

Proof

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