关于自由Stein维数
On Free Stein Dimension
摘要：
我们建立了自由Stein维数的几个性质，自由Stein维数是一个不变量，它是一个可生成的单位迹$*$ -代数的不变量。我们给出了它在有限维代数的直和和张量积下的性质公式。在一组给定的发电机，我们表明，（近似）代数关系产生（非近似）界上的自由斯坦维。特别治疗的情况下，可分离的阿贝尔冯诺依曼代数，我们表明，自由Stein维数是冯诺依曼代数不变量。此外，我们表明，在温和的假设下，$L^2$刚性意味着自由斯坦维1。最后，我们使用限制上/下扩展自由Stein维的冯诺依曼代数不变量-这是实质上更难以计算一般-并计算它在几种情况下的利益。
英文摘要：
We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a given set of generators, we show that (approximate) algebraic relations produce (non-approximate) bounds on the free Stein dimension. Particular treatment is given to the case of separable abelian von Neumann algebras, where we show that free Stein dimension is a von Neumann algebra invariant. In addition, we show that under mild assumptions $L^2$-rigidity implies free Stein dimension one. Finally, we use limits superior/inferior to extend the free Stein dimension to a von Neumann algebra invariant -- which is substantially more difficult to compute in general -- and compute it in several cases of interest.
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