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Spectral permanence for unital c star subalgebras

Statement

Assume AC. If $B$ is a unital C*-subalgebra of a unital C*-algebra $A$ with the same identity, then $σ_{B} (b) = σ_{A} (b)$ for every $b \in B$ .

Facts & Assumptions

[A1]

A unital C*-algebra is a nonzero complex Banach algebra with a unit of norm one and an involution satisfying the C*-identity; a unital C*-subalgebra with the same identity is a closed unital $*$ -subalgebra containing that unit (C star algebra, Unital Banach algebra).

[A2]

For every element $b$ invertible in $A$ the adjoint $b^{*}$ is invertible with $(b^{*})^{- 1} = (b^{- 1})^{*}$ , and $b^{*} b$ is positive; positivity of an element means $x = y^{*} y$ for some element $y$ (Self-adjoint positive unitary and normal elements for positivity, C star algebra for the involution rules).

[A3]

If $B \subseteq A$ is a unital subalgebra with the same unit and $b \in B$ , then $σ_{A} (b) \subseteq σ_{B} (b)$ , since invertibility in $B$ implies invertibility in $A$ (Spectrum and resolvent set in a Banach algebra).

[A4]

For a nonzero unital commutative complex C*-algebra $C$ the Gelfand transform $Γ : C \to C (Δ (C))$ is an isometric unital $*$ -isomorphism onto $C (Δ (C))$ , and $Δ (C)$ is a nonempty compact Hausdorff space (Commutative Gelfand Naimark, Maximal ideal space is compact Hausdorff, Character and maximal ideal space).

[A5]

A unital self-adjoint complex function algebra that separates the points of a nonempty compact Hausdorff space is uniformly dense in all continuous complex functions (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[A6]

Characters of a unital commutative complex C*-algebra satisfy $χ (c^{*}) = \overline{χ (c)}$ (Characters on a unital commutative C star algebra preserve star).

[A7]

AC is the global hypothesis of the Gelfand-theoretic suppliers (The Axiom of Choice).

Proof

technique · direct

Given: A unital C*-algebra $A$ , a unital C*-subalgebra $B \subseteq A$ with $1_{B} = 1_{A}$ , and $b \in B$ .

1.1A3

Since invertibility in $B$ implies invertibility in $A$ , one has $σ_{A} (b) \subseteq σ_{B} (b)$ ; it remains to prove the reverse inclusion, that is, that $b$ invertible in $A$ is invertible in $B$ with inverse in $B$ .

1.2A1A2algebra

If $b$ is invertible in $A$ , then $x : = b^{*} b \in B$ is positive and invertible in $A$ with $x^{- 1} = b^{- 1} (b^{- 1})^{*}$ ; and if $x$ positive and invertible in $A$ has $x^{- 1} \in B$ , then $(b^{*} b)^{- 1} \in B$ and $b^{- 1} = (b^{*} b)^{- 1} b^{*} \in B$ .

1.3A1A2A4A7algebra

It suffices to treat a self-adjoint $x \in B$ invertible in $A$ , because the positive element $b^{*} b$ in the preceding reduction is self-adjoint. For such $x$ , let $C : = C_{A}^{*} (1, x, x^{- 1})$ be the closed unital star-subalgebra of $A$ generated by $x$ and $x^{- 1}$ , and let $D : = C_{A}^{*} (1, x) \subseteq B$ . Both are commutative, because $x = x^{*}$ and $(x^{- 1})^{*} = x^{- 1}$ , and the Gelfand transform is an isometric unital star-isomorphism $Γ : C \to C (Δ (C))$ on the nonempty compact Hausdorff character space.

2.1step 1.3A6algebra

The function $\hat{x}$ separates the points of $Δ (C)$ . Indeed, if characters $h, h^{'}$ have $h (x) = h^{'} (x)$ , then $h (x^{- 1}) = h (x)^{- 1} = h^{'} (x)^{- 1} = h^{'} (x^{- 1})$ ; they therefore agree on the unital star-algebra generated by $x, x^{- 1}$ and, by continuity, on its closure $C$ , so $h = h^{'}$ . Moreover $\hat{x}$ is real-valued because $x = x^{*}$ and characters preserve the involution.

3.1step 2.1A4A5

Under $Γ$ , the algebra $D$ is the closed unital self-adjoint function algebra generated by $\hat{x}$ . It separates points by step 2.1, so Stone--Weierstrass gives $Γ (D) = C (Δ (C)) = Γ (C)$ . Injectivity of $Γ$ yields $D = C$ , hence $x^{- 1} \in D \subseteq B$ .

4.1step 1.1step 1.2step 3.1∎

Therefore every $b \in B$ invertible in $A$ has $(b^{*} b)^{- 1} \in B$ by step 3.1 and then $b^{- 1} = (b^{*} b)^{- 1} b^{*} \in B$ by step 1.2. Thus $σ_{B} (b) \subseteq σ_{A} (b)$ ; with step 1.1 this gives $σ_{B} (b) = σ_{A} (b)$ .

Spectral permanence for unital c star subalgebras

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources