On a canonical lattice structure on the effect algebra of a von Neumann algebra
On a canonical lattice structure on the effect algebra of a von Neumann algebra
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冯诺依曼代数效应代数的正则格结构
DOI：
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发表时间：
2004
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通讯作者：
H. F. D. Groote
中科院分区：
文献类型：
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作者：
H. F. D. Groote
设R是作用在Hilbert空间H上的von Neumann代数，Rsa是R的自伴元集。众所周知，Rsa是关于通常的偏序≤的格当且仅当R是交换的。定义并研究了R_sa上的一种新的偏序--谱序≤_S，它推广了≤在投影上的性质，它比通常的谱序粗糙，但与交换子代数上的谱序一致，使R_sa成为有界完备格.当且仅当R是有限von Neumann代数时，效应代数E(R)：={A|0≤A≤I}是完备格，且证明了映射A-->；R(A)是从格E(R)到投影格P(R)的同态.
Let R be a von Neumann algebra acting on a Hilbert space H and let R_sa be the set of selfadjoint elements of R. It is well known that R_sa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on R_sa, the spectral order ≤_s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on abelian subalgebras, and turns R_sa into a boundedly complete lattice. The effect algebra E(R) := {A | 0 ≤ A ≤ I} is then a complete lattice and we show that the mapping A --> R(A), where R(A) denotes the range projection of A, is a homomorphism from the lattice E(R) onto the projection lattice P(R) of A if and only if R is a finite von Neumann algebra.