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Chapter 1
- Cuntz-Krieger family and the Cuntz-Krieger relations
- Since projections \(P_{v}\) are mutually orthogonal, the ranges \(P_{v}\mathcal{H}\) are mutually orthogonal subspaces of \(\mathcal{H}\).
- The first CK relation says \(S_{e}\) has initial space \(P_{s(e)}\mathcal{H}\)
- Second relation implies that the range projection \(S_{e}S^*_{e}\) is dominated by \(P_{r(e)}\), hence \(S_{e}\mathcal{H} \subset P_{r(e)}\mathcal{H}\).
- \(S_{e}\) is an isometry of \(P_{s(e)}\mathcal{H}\) onto a closed subspace of \(P_{r(e)}\mathcal{H}\). This gives
\[ S_{e} = P_{r(e)}S_{e} = S_{e}P_{s(e)} \] which is very important function.
- Further, CK2 implies that \(S_{e}\) for each edge \(e\) with \(r(e) = v\) have mutually orthogonal ranges with span \(P_{v}\mathcal{H}\) so
\[ P_{v}\mathcal{H} = \bigoplus_{\{ e \in E^1 : r(e) = v \}} S_{e}\mathcal{H} \]
Finding Cuntz-Krieger $E$-families with \(P_{v}, S_{e}\) nonzero
Take \(H_{v}\) to be a separable infinite-dimensional Hilbert space for each \(v \in E^0\), set \(\mathcal{H} = \bigoplus_{v} \mathcal{H}_{v}\). Then take \(P_{v}\) to be the projection of \(\mathcal{H}\) on \(\mathcal{H}_{v}\), decompose \(\mathcal{H}_{v}\) as a direct sum \(\mathcal{H}_{v} = \bigoplus_{r(e) = v} H_{v,e}\)
Examples
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