Kernels of conditional determinantal measures

For determinantal point processes governed by self-adjoint kernels, we prove in Theorem 1.2 that conditioning on the configuration in a subset preserves the determinantal property. In Theorem 1.3 we show the tail sigma-algebra for our determinantal point processes is trivial, proving a conjecture by Lyons. If our self-adjoint kernel is a projection, then, establishing a conjecture by Lyons and Peres, we show in Theorem 1.5 that reproducing kernels corresponding to particles of almost every configuration generate the range of the projection. Our argument is based on a new local property for conditional kernels of determinantal point processes stated in Lemma 1.7