Manuscript XVII-A@1 · XVII-A.tex · lemma · 711d7ee4

Lemma: Every open time interval is algebraically complete

Boundary data and the declared representation correspondence › A finite-cutoff localization caveat · lines 418–482 · proof included or linked

XVII-A@1#lem:timeband, XVII-A@1#eq:full-algebra

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For the finite model,

A (B) = B (H_{\partial}) .

The same conclusion holds when $I_{B}^{\circ}$ is replaced by any nonempty open time interval.

Proof. Write

C_{n} = b_{n} + b_{n}^{†}, T_{n} = i (b_{n}^{†} - b_{n}) .

Then

β (t) = n = 1 \sum^{K} c_{n} (C_{n} cos (n t) + T_{n} sin (n t)) .

The $2 K$ real functions

u_{α} (t) \in {cos (n t), sin (n t) : 1 \leq n \leq K}

are linearly independent on every nonempty open interval. Indeed, a trigonometric polynomial vanishing there is identically zero, and its distinct Fourier coefficients then vanish.

Choose $w \in C_{c}^{\infty} (I_{B}^{\circ})$ , $w \geq 0$ , positive on some nonempty open subinterval. The real Gram matrix

G_{α γ} = \int w (t) u_{α} (t) u_{γ} (t) d t

is positive definite. The test functions

f_{α} (t) = w (t) γ \sum (G^{- 1})_{α γ} u_{γ} (t)

satisfy

\int f_{α} (t) u_{η} (t) d t = δ_{α η} .

The smearings $β (f_{α})$ therefore isolate $c_{n} C_{n}$ and $c_{n} T_{n}$ . Since $c_{n} \neq = 0$ , the algebra contains

b_{n} = \frac{1}{2} (C_{n} + i T_{n}) and b_{n}^{†} .

On one factor, $b_{n}^{†} b_{n}$ has the distinct eigenvalues $0, \dots, d - 1$ . Polynomial functional calculus supplies all its one-dimensional spectral projections $P_{m}$ . Moreover,

P_{m - 1} b_{n} P_{m} = m ∣ m - 1 ⟩ ⟨ m ∣ .

Taking adjoints and products supplies every matrix unit on that factor. The factor algebras together generate $B ((C^{d})^{\otimes K})$ , proving the assertion.

Displayed formulas in this passage

XVII-A@1#eq:full-algebra · line 421
$A (B) = B (H_{\partial}) .$
unlabeled · line 431
$C_{n} = b_{n} + b_{n}^{†}, T_{n} = i (b_{n}^{†} - b_{n}) .$
unlabeled · line 436
$β (t) = n = 1 \sum^{K} c_{n} (C_{n} cos (n t) + T_{n} sin (n t)) .$
unlabeled · line 442
$u_{α} (t) \in {cos (n t), sin (n t) : 1 \leq n \leq K}$
unlabeled · line 452
$G_{α γ} = \int w (t) u_{α} (t) u_{γ} (t) d t$
unlabeled · line 457
$f_{α} (t) = w (t) γ \sum (G^{- 1})_{α γ} u_{γ} (t)$
unlabeled · line 463
$\int f_{α} (t) u_{η} (t) d t = δ_{α η} .$
unlabeled · line 468
$b_{n} = \frac{1}{2} (C_{n} + i T_{n}) and b_{n}^{†} .$
unlabeled · line 476
$P_{m - 1} b_{n} P_{m} = m ∣ m - 1 ⟩ ⟨ m ∣ .$

Lemma: Every open time interval is algebraically complete

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