三类正规序列效应代数
The three types of normal sequential effect algebras
摘要：
序贯效应代数（英语：Sequential Effect Algebra，SEA）是一种效应代数，它具有以C*-代数上的Lüders乘积$(a,b)\mapsto \sqrt{a}b\sqrt{a}$为模型的序贯乘积运算。当SEA具有所有有向集的上确界，并且序列积与这些上确界适当地相互作用时，SEA被称为正规的。对希尔伯特空间和冯·诺依曼或JBW代数的单位区间的影响是标准SEA的例子，这些SEA是额外凸的，即拥有代数上的实单位区间的适当作用。完全布尔代数也形成正规的SEA，它只有当$0=1$是凸的。我们表明，任何正常的海$E$分裂为一个完整的布尔代数$E_b$，凸正常海PDMATHAMR，和一个新确定的类型的正常海$E_{ac}$，我们纯粹的几乎凸的直和$E\equiv E_b\oplus E_c \oplus E_{ac}$。一路上，我们表明，除其他事项外，一个海只包含幂等元必须是一个布尔代数，我们建立了一个谱定理，使用它我们解决了一类正常的海的一个问题古德关于平方根的唯一性。在建立了我们的主要结果之后，我们提出了一个简单的额外公理，排除了看似病态的a-凸的SEA。最后，我们通过研究与关联序列产品的SEAs的文件。我们发现，结合性迫使正常的SEAs满足我们的新公理是交换的，揭示了为什么在量子理论中的序列产品应该是非结合的问题。
英文摘要：
A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product $(a,b)\mapsto \sqrt{a}b\sqrt{a}$ on C*-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when $0=1$. We show that any normal SEA $E$ splits as a direct sum $E\equiv E_b\oplus E_c \oplus E_{ac}$ of a complete Boolean algebra $E_b$, a convex normal SEA $E_c$, and a newly identified type of normal SEA $E_{ac}$ we dub purely almost-convex. Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.
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