论文标题
W* - 代数为von-Neumann代数的完整公理化
W*-Algebras as a complete axiomatisation of Von-Neumann algebras
论文作者
论文摘要
通过Gelfand-Naimark定理，任何C *-Algebra都是同构对Hilbert空间上有界操作员的 *代数的同构，该空间是根据操作员规范引起的拓扑结构而封闭的。因此，C* - 代数提供了这些操作员代数的结构的公理化。实际上，在封闭 * - 代数上考虑的整个结构信息 - 即添加，标量乘法，操作员的相邻和组成以及运算符的标准 - 在C *-Algebra中反映了公理化的整个结构信息。从这个角度来看，我们对待von-Neumann代数，另一类运营商代数也定义为封闭的 * - 代数，但就另一种拓扑而言，一种可能性是弱操作员拓扑。 S. sakai解决了von-neumann代数的公理化问题，S. sakai引入了所谓的W* - 代数。它们是C* - 代数，被认为是Banach空间，对另一个称为Predual的Banach空间的双重空间是同构的。 Sakai的定理断言W* - 代数提供了Von-Neumann代数的公理化。在这本硕士论文中，我们给出了Sakai定理的证明。实际上，我们提供了更强的Sakai定理版本，该版本表明W* - 代数确实形成了类似于C*-Algebras情况的完整的公理化。此外，我们显示出对等轴测异构形态的独特性，通过使用我们加强的Sakai定理，简化了Sakai的原始证据。
By the Gelfand-Naimark theorem, any C*-algebra is isometrically isomorphic to a *-algebra of bounded operators on a Hilbert space which is closed with respect to the topology induced by the operator norm. Hence, the C*-algebras furnish an axiomatisation of the structure of those operator algebras. In fact, the axiomatisation is complete in the sense that the entire structural information considered on the closed *-algebra -- namely addition, scalar multiplication, adjunction and composition of operators as well as the operator norm -- is reflected in the C*-algebra. From this point of view, we treat Von-Neumann algebras, another class of operator algebras also defined as closed *-algebras, but with respect to another topology -- one possibility is the weak operator topology. The problem asking for an axiomatisation of Von-Neumann algebras was solved by S. Sakai who introduced so-called W*-algebras. They are C*-algebras which, considered as a Banach space, are isometrically isomorphic to the dual space of another Banach space called the predual. Sakai's theorem asserts that the W*-algebras provide an axiomatisation of Von-Neumann algebras. In this master's thesis, we give a proof of Sakai's theorem. In fact, we provide a stronger version of Sakai's theorem which shows that W*-algebras indeed form a complete axiomatisation analogous to the situation for C*-algebras. Additionally, we show uniqueness of the predual up to isometric isomorphism, simplifying Sakai's original proof by using our strengthened version of Sakai's theorem.
