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

Boundary operator family

Boundary data and the declared representation correspondence › Boundary operator family · lines 330–350

XVII-A@1#eq:boundary-field, XVII-A@1#eq:boundary-algebra

On the boundary copy define

β (t) = n = 1 \sum^{K} c_{n} (b_{n} e^{- in t} + b_{n}^{†} e^{in t}), c_{n} = (- 1)^{n + 1} \frac{n}{π} .

Let $A (B)$ be the unital $*$ -algebra generated by

β (f) = \int_{I_{B}} f (t) β (t) d t, f \in C_{c}^{\infty} (I_{B}^{\circ}; R),

inside $B (H_{\partial})$ . In finite dimension this algebra is norm closed.

Displayed formulas in this passage

XVII-A@1#eq:boundary-field · line 333
$β (t) = n = 1 \sum^{K} c_{n} (b_{n} e^{- in t} + b_{n}^{†} e^{in t}), c_{n} = (- 1)^{n + 1} \frac{n}{π} .$
XVII-A@1#eq:boundary-algebra · line 342
$β (f) = \int_{I_{B}} f (t) β (t) d t, f \in C_{c}^{\infty} (I_{B}^{\circ}; R),$