II-1 subfactor

In the theory of von Neumann algebras, a subfactor of a factor ${\displaystyle M}$ is a subalgebra that is a factor and contains ${\displaystyle 1}$. The theory of subfactors led to the discovery of the Jones polynomial in knot theory.

Usually ${\displaystyle M}$ is taken to be a factor of type ${\displaystyle {\rm {II}}_{1}}$, so that it has a finite trace. In this case every Hilbert space module ${\displaystyle H}$ has a dimension ${\displaystyle \dim _{M}(H)}$ which is a non-negative real number or ${\displaystyle +\infty }$. The index ${\displaystyle [M:N]}$ of a subfactor ${\displaystyle N}$ is defined to be ${\displaystyle \dim _{N}(L^{2}(M))}$. Here ${\displaystyle L^{2}(M)}$ is the representation of ${\displaystyle N}$ obtained from the GNS construction of the trace of ${\displaystyle M}$.

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