Strongly 1-Bounded Von Neumann Algebras
Strongly 1-Bounded Von Neumann Algebras
复制标题
强 1 有界冯诺依曼代数
DOI：
10.1007/s00039-007-0624-9
复制
发表时间：
2005
影响因子：
2.2
通讯作者：
Kenley Jung
中科院分区：
文献类型：
--
作者：
Kenley Jung
摘要。假设F是迹冯诺伊曼代数M中自伴元素的有限元组。对于α > 0，F是α-有界的，如果
$${\mathbb{P}}^\alpha（F）< \infty$$
哪里
$${\mathbb{P}}^\alpha$$
是[J3]中引入的F的自由堆积α-熵。称M是强1-有界的，如果M有一个1-有界的自伴生成元有限元组F，使得存在一个
$$x \in F$$
与
$$\chi（x）> -\infty$$
.证明了：若M是强1-有界的，则M的任何自伴生成元G的有限元组都是1-有界的且δ0（G）≤ 1，从而证明了强1-有界的von Neumann代数不同构于内插自由群因子，且δ0是这类代数的不变量.强1-有界von Neumann代数的例子包括（可分）II 1-因子，它们具有性质Γ，具有Cartan子代数，是非素的，或
$$SL_n（{\mathbb{Z}}），n \geq 3$$
.若M和N是强1-有界的，且M <$N是扩散的，则由M和N生成的von Neumann代数是强1-有界的。特别地，两个强1-有界冯诺依曼代数在一个公共的扩散冯诺依曼子代数上合并的自由积是强1-有界的。证明了由强1-有界von Neumann子代数的正规化子生成的II 1-因子是强1-有界的。
Abstract.Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if
$${\mathbb{P}}^\alpha (F) < \infty$$
where
$${\mathbb{P}}^\alpha$$
is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an
$$x \in F$$
with
$$\chi (x) > -\infty$$
. It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ0(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ0 is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II1-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of
$$SL_n({\mathbb{Z}}), n \geq 3$$
. If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.