Factor algebra

An involutive subalgebra $\mathfrak A$ of the algebra $B(H,H)$ of linear operators on a Hilbert space $H$ that is closed relative to so-called weak convergence of operators and has the property that its centre (that is, the collection of all operators in $\mathfrak A$ that commute with every operator in $\mathfrak A$) consists of scalar multiples of the unit operator.

If $\mathfrak A$ is a factor, then for a large supply of subspaces $F$ of $H$ one can define the concept of the dimension $\dim_\mathfrak AF$ relative to $\mathfrak A$ as an invariant that is preserved, not under arbitrary isometries $\mathcal F$, but only under those in the given factor with additional natural properties (for example, $\dim_\mathfrak A(F_1\oplus F_2)=\dim_\mathfrak AF_1+\dim_\mathfrak AF_2$). All factors can be divided into five classes corresponding to the values that $\dim_\mathfrak AF$ can take, where, for example, for a factor of class $\Pi_\infty$ it can take any value in $[0,\infty]$.

Comments

An involutive algebra is an algebra over $\mathbf C$ endowed with an involution. For information concerning various types of factors cf. von Neumann algebra.

References

[a1]	G.K. Pedersen, "$C^*$-algebras and their automorphism groups" , Acad. Press (1979)
[a2]	J. Dixmier, "$C^*$ algebras" , North-Holland (1977) (Translated from French)