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*-A bicommutant is its own bicommutant.
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bicommutant and is
Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M ′′ of M. This algebra is the von Neumann algebra generated by M.
In algebra, the bicommutant of a subset S of a semigroup ( such as an algebra or a group ) is the commutant of the commutant of that subset.
The bicommutant is particularly useful in operator theory, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebras.
Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C *- algebra B ( H ), for some Hilbert space H, then the weak closure, strong closure and bicommutant of M are equal.
So, i. e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
# the effect of the operations is to yield an interesting bicommutant theory.
In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors ; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level.
bicommutant and .
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set.
The bicommutant of S always contains S. So.
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