There are two subjects in which non-integral dimensions appear:
- fractal geometry: consider the well-known Hausdorff dimension of fractals.
- von Neumann algebra: consider a type ${\rm II_1}$ factor $M$ and a $M$-module $H$, then von Neumann defined a notion of $M$-dimension $\dim_M(H) \in [0, \infty]$. Moreover, for any $\alpha \in [0, \infty]$ there is a $M$-module $H_{\alpha}$ with $\dim_M(H_{\alpha}) = \alpha$ (and two $M$-modules of same $M$-dimension are isomorphic).
Question: Is there a link between Hausdorff dimension and von Neumann dimension ?
More precisely, from a given fractal $\mathcal{F}$ of Hausdorff dimension $\alpha$, can we make a type ${\rm II_1}$ factor $M$ and a $M$-module $H$ such that $\dim_M(H) = \alpha$ ?
Or conversely, from a given type ${\rm II_1}$ factor $M$ and $M$-module $H$ with $\dim_M(H) = \alpha$, can we naturally make a fractal of Hausdorff dimension $\alpha$? Should we intuitively think about a connected or a totally disconnected fractal (like a Cantor set)? If $\alpha \le 2$, is there a natural way to draw it?
Remark: such a link already exists between Hausdorff dimension of fractals and dimension spectrum of Connes' spectral triples (see "Fractals in Noncommutative Geometry" by Guido-Isola arXiv:math/0102209, MR1867554, inspired by Chapter 4, Section 3 of Connes' book).
Remark: there also exist notions of (non-integral) quantum dimension, statistical dimension, Perron-Frobenius dimension (of an object in a (unitary) fusion category), or subfactor index, but (most of the time) they can be formulated in term of a von Neumann dimension.
