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II_{\infty} --> II_1 + converse of the precision and additional questions

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edited May 21, 2019 at 18:37

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_{\infty}}$${\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_{\infty}}$${\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.

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 ${\rm II_{\infty}}$ 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 ${\rm II_{\infty}}$ factor $M$ and a $M$-module $H$ such that $\dim_M(H) = \alpha$ ?

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?

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asked May 12, 2019 at 13:29

Sebastien Palcoux

Hausdorff dimension and von Neumann dimension

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 ${\rm II_{\infty}}$ 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 ${\rm II_{\infty}}$ factor $M$ and a $M$-module $H$ such that $\dim_M(H) = \alpha$ ?