Type Iₙ factor - Imported Knowlpedia Content

Definition

Let $n\geq 1$ be an integer. A type $\mathrm{I}_n$ factor is a type I factor isomorphic, as a von Neumann algebra, to the matrix algebra $M_n(\mathbb C)$ . Equivalently, it is a type I factor whose maximal families of nonzero mutually orthogonal equivalent minimal projections have exactly $n$ members. Its identity is finite, so it is a finite von Neumann algebra. The integer $n$ is intrinsic: it is the complex dimension of the Hilbert space $K$ in the realization $M\cong B(K)$ , not the vector-space dimension $n^2$ of the algebra.

Matrix-unit description

A type $\mathrm{I}_n$ factor contains matrix units $\{e_{ij}\}_{1\leq i,j\leq n}$ satisfying

e_{ij}e_{kl}=\delta_{jk}e_{il},\qquad e_{ij}^*=e_{ji},\qquad \sum_{i=1}^n e_{ii}=1.

The diagonal projections $e_{ii}$ are mutually equivalent and minimal. Conversely, such a full system of matrix units identifies the algebra with $M_n(\mathbb C)$ . The classification follows from the type I factor theorem Takesaki, Chapter V, §1.

Trace and examples

There is a unique normalized tracial state,

\tau(x)=\frac{1}{n}\operatorname{Tr}(x).

For projections, $\tau(p)=\operatorname{rank}(p)/n$ , so the possible trace values form the finite set $\{0,1/n,\ldots,1\}$ .

The case $n=1$ is $\mathbb C$ . The case $n=2$ is the algebra of $2\times2$ complex matrices. An infinite type I factor $B(H)$ is not type $\mathrm{I}_n$ for any finite $n$ , even though it contains many corners isomorphic to finite matrix algebras.

Conventions and scope

References

M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §1 on type I factors and their dimension.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: §6.5 on finite type I factors.