Factor algebra
An involutive subalgebra of the algebra
of linear operators on a Hilbert space
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
that commute with every operator in
) consists of scalar multiples of the unit operator.
If is a factor, then for a large supply of subspaces
of
one can define the concept of the dimension
relative to
as an invariant that is preserved, not under arbitrary isometries
, but only under those in the given factor with additional natural properties (for example,
). All factors can be divided into five classes corresponding to the values that
can take, where, for example, for a factor of class
it can take any value in
.
Comments
An involutive algebra is an algebra over endowed with an involution. For information concerning various types of factors cf. von Neumann algebra.
References
[a1] | G.K. Pedersen, "![]() |
[a2] | J. Dixmier, "![]() |
Factor algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factor_algebra&oldid=11524