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000065 | \begin{definition}[One-sided boundary admissibility datum]
000066 | \label{def:reduced-theorem-datum}
000067 | The one-sided boundary admissibility datum is
000068 | \[
000069 | \Tbound:=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma)
000070 | \]
000071 | with the following built-in admissibility conditions:
000072 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000073 |     \item $Z(\mathcal A_+)$ is spectrally regular and its Gelfand spectrum is a smooth compact oriented manifold with boundary $M_+$;
000074 |     \item $B_\Sigma$ is a self-adjoint tangential boundary operator on $\Sigma:=\partial M_+$ compatible with the boundary restrictions of $J$ and $\Gamma$;
000075 |     \item in a collar neighborhood of $\Sigma$, the Dirac operator has canonical product form
000076 |     \[
000077 |     D_+=\gamma_n(\partial_n+B_\Sigma);
000078 |     \]
000079 |     \item the boundary collar is reflection-regular so that canonical doubling, the Calder\'on projector, and the admissibility complex are well posed on the same datum.
000080 | \end{enumerate}
000081 | \end{definition}
000082 | 
000083 | \begin{remark}[Almost-commutative realizations]
000084 | \label{ass:almost-commutative-factorization}
000085 | When a factorized realization is useful, one may choose
000086 | \[
000087 | \mathcal H\cong L^2(\widetilde M,S)\otimes \Hint,
000088 | \]
000089 | with $S\to\widetilde M$ the spinor bundle on the oriented double cover and $\Hint$ a finite
000090 | internal Hilbert space. This is an analytic realization choice rather than part of the boundary
000091 | datum.
000092 | \end{remark}