There are two subjects in which non-integral dimensions appear:     
 
- [fractal geometry][1]:  consider the well-known [Hausdorff dimension of fractals][2].  
- [von Neumann algebra][3]: consider a type ${\rm II_{\infty}}$ factor $M$ and a [$M$-module][4] $H$, 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 ${\rm II_{\infty}}$ factor $M$ and a $M$-module $H$ such that $\dim_M(H) = \alpha$ ?  

*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][7], [MR1867554][6], inspired by Chapter 4, Section 3 of [Connes' book][5]).  

*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.

  [1]: https://en.wikipedia.org/wiki/Fractal
  [2]: https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
  [3]: https://en.wikipedia.org/wiki/Von_Neumann_algebra
  [4]: https://en.wikipedia.org/wiki/Von_Neumann_algebra#Modules_over_a_factor
  [5]: http://www.alainconnes.org/docs/book94bigpdf.pdf
  [6]: https://mathscinet.ams.org/mathscinet-getitem?mr=1867554
  [7]: https://arxiv.org/abs/math/0102209