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Local spin factor inclusions preserve the operator norm

Statement

Fix $d \geq 1$ , $q \geq 1$ , and finite regions $Λ \subseteq Λ^{'} \subseteq Z^{d}$ as in Local spin matrix algebras and their factor inclusions. Then the factor inclusion $ι_{Λ, Λ^{'}} (T) = T \otimes I$ is an isometric unital $*$ -homomorphism:

$∥ ι_{Λ, Λ^{'}} (T) ∥ = ∥ T ∥ (T \in A_{Λ}) .$

For nested finite regions $Λ \subseteq Λ^{'} \subseteq Λ^{''}$ , the inclusions compose: $ι_{Λ^{'}, Λ^{''}} \circ ι_{Λ, Λ^{'}} = ι_{Λ, Λ^{''}}$ .

Facts & Assumptions

Given: $d \geq 1$ , $q \geq 1$ , finite $Λ \subseteq Λ^{'} \subseteq Λ^{''} \subseteq Z^{d}$ , and the configuration Hilbert spaces and inclusions of Local spin matrix algebras and their factor inclusions.

[F1]

For finite $Λ$ , $H_{Λ} = ℓ^{2} ({1, \dots, q}^{Λ})$ ; for $Λ \subseteq Λ^{'}$ , restriction of configurations identifies $H_{Λ^{'}}$ with $H_{Λ} \otimes H_{Λ^{'} ∖ Λ}$ , and $ι_{Λ, Λ^{'}} (T) = T \otimes I$ (Local spin matrix algebras and their factor inclusions).

[F2]

Each finite configuration Hilbert space has a finite orthonormal configuration basis, and $∥ T ∥ = \sup_{∥ u ∥ = 1} ∥ T u ∥$ is its induced operator norm (Local spin matrix algebras and their factor inclusions).

Proof

technique · direct

Given: $d \geq 1$ , $q \geq 1$ , the finite regions and configuration Hilbert spaces of the Statement.

1.1givenF1algebra

Under [F1], $T \mapsto T \otimes I$ preserves products and adjoints because $(T \otimes I) (S \otimes I) = T S \otimes I$ and $(T \otimes I)^{*} = T^{*} \otimes I$ ; it sends $I_{Λ}$ to $I_{Λ^{'}}$ . The configuration-basis factorization is associative, so for nested regions the two successive inclusions and the direct inclusion act identically on every configuration basis vector and hence are equal operators.

1.2givenF1F2algebra

Let $K = H_{Λ^{'} ∖ Λ}$ , and choose its finite orthonormal configuration basis $e_{1}, \dots, e_{s}$ . Every $ξ \in H_{Λ} \otimes K$ has a unique expansion $ξ = \sum_{j = 1}^{s} ξ_{j} \otimes e_{j}$ , with $∥ ξ ∥^{2} = \sum_{j} ∥ ξ_{j} ∥^{2}$ . Therefore $∥ (T \otimes I) ξ ∥^{2} = \sum_{j} ∥ T ξ_{j} ∥^{2} \leq ∥ T ∥^{2} ∥ ξ ∥^{2}$ , so $∥ T \otimes I ∥ \leq ∥ T ∥$ .

1.3givenF1F2constructalgebra

If $T = 0$ , both norms are zero. Otherwise, for every $ε > 0$ the supremum definition in [F2] gives a unit vector $u \in H_{Λ}$ with $∥ T u ∥ > ∥ T ∥ - ε$ . Since $q \geq 1$ , $K$ is nonzero; choose a unit configuration vector $v \in K$ . Then $∥ (T \otimes I) (u \otimes v) ∥ = ∥ T u ∥ ∥ v ∥ > ∥ T ∥ - ε$ , so $∥ T \otimes I ∥ \geq ∥ T ∥ - ε$ for every $ε > 0$ .

2.1step 1.1step 1.2step 1.3∎

Letting $ε ↓ 0$ in step 1.3 and combining with step 1.2 gives $∥ ι_{Λ, Λ^{'}} (T) ∥ = ∥ T ∥$ . Step 1.1 supplies the unital $*$ -homomorphism and composition identities, completing the claim.

Depends on

Local spin matrix algebras and their factor inclusions

Local spin factor inclusions preserve the operator norm

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources