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SWUFE数学讲坛103：部分分解性质与算子代数相应的不变子空间

In this talk, we will consider partial factorization properties in operator algebras. Let \mathbf{M} be a von Neumann algebra and A \subseteq M a unital subalgebra. If for any invertible operator S\in\mathbf{M}, there exists an isometry(resp. a co-isometry) U\in\mathbf{M} such that U^\astS,\ S^{-1}U\in\mathbf{A}, then we say that A has the left(resp. right) partial factorization property. If U is unitary, then we say that A has the factorization property. We prove that the relative invariant subspace lattice of a unital subalgebra A with the left(resp. right) partial factorization property in the von Neumann algebra \mathbf{M} is commutative. Moreover, if the Hilbert space is separable, then this lattice is generated by a nest and the projections in the center of \mathbf{M}. In particular, we concern these properties for subdiagonal algebras.