Dimension function on projections - Imported Knowlpedia Content

Definition

Let $M$ be a factor with separable predual and let $\mathcal P(M)$ be its projections. A dimension function on projections is a map

d:\mathcal P(M)\longrightarrow[0,\infty]

that is invariant under Murray–von Neumann equivalence, additive on orthogonal projections, faithful $(d(p)=0\Rightarrow p=0)$ , and normal:

p_i\uparrow p\quad\Longrightarrow\quad d(p_i)\uparrow d(p).

In a finite factor one requires $d(1)=1$ . In an infinite semifinite factor a choice of scale is required; in a type I factor it is customary to give every minimal projection dimension $1$ . These normalizations make dimension an extension of matrix rank or normalized rank.

Dimension scales and factor types

With the standard normalization, the possible finite values distinguish the factor types. They are $\{0,1/n,\ldots,1\}$ for type $\mathrm I_n$ , $[0,1]$ for type $\mathrm{II}_1$ , $\{0,1,2,\ldots\}$ for type $\mathrm I_\infty$ , and $[0,\infty)$ for type $\mathrm{II}_\infty$ . For a type III factor the only scalar dimension function has values $0$ and $\infty$ . This is the continuous-dimension theory underlying the classification of factors Takesaki, Chapter V, §§1–2.

Finite factors and traces

If $M$ is finite, its unique normalized center-valued trace is scalar because $M$ is a factor. Its restriction to projections is the normalized dimension function. In a type $\mathrm{II}_1$ factor, this gives

p\sim q\quad\Longleftrightarrow\quad d(p)=d(q),

and every value in $[0,1]$ occurs. Thus dimension is a complete invariant of projection equivalence in this setting, not merely a numerical estimate.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §§1–2 on equivalence, comparison, dimension, and factor types.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: Chapter 6 on comparison theory and dimension of projections.