论文标题
在免费的斯坦因维度上
On Free Stein Dimension
论文作者
论文摘要
我们建立了自由Stein Dimension的几个属性，这是有限生成的Unital Tracial $*$ - 代数不变的。我们在具有有限维代数的直接总和和张量产品下为其行为提供了公式。在给定的一组发电机中，我们表明（近似）代数关系会在自由施坦尺寸上产生（非同胞）界限。对可分离的Abelian von Neumann代数的情况进行了特殊处理，在这里我们表明自由Stein Dimension是Von Neumann代数不变的。此外，我们表明，在轻度假设下，$ l^2 $ rigities意味着免费的Stein尺寸。最后，我们使用上等/下等极限将自由Stein尺寸扩展到von Neumann代数不变的不变性 - 通常要计算得更难以计算 - 并在一些感兴趣的情况下对其进行计算。
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.
