commutant is a weak operator closed subalgebra


Let H be a Hilbert spaceMathworldPlanetmath and B(H) the algebra of bounded operatorsMathworldPlanetmathPlanetmath in H. Recall that the commutant of a subset B(H) is the set of all bounded operators that commute with those of , i.e.

:={TB(H):TS=ST,S}.

- If B(H), then is a subalgebra of B(H) that contains the identity operatorMathworldPlanetmath and is closed in the weak operator topology.

: It is clear that contains the identity operator, since it commutes with all operatorsMathworldPlanetmath in B(H) and in particular with those of .

Let us now see that is a subalgebra of B(H). Let T1,T2 and λ. We have that, for all S,

S(T1+T2)=ST1+ST2=T1S+T2S=(T1+T2)S
S(λT1)=λST1=λT1S
S(T1T2)=T1ST2=T1T2S

thus, T1+T2, λT1 and T1T2 all belong to , and therefore is a subalgebra of B(H).

It remains to see that is weak operator closed. Suppose (Ti) is a net in that convergesPlanetmathPlanetmath to T in the weak operator topology. Then, for all x,yH we have that Tix,yTx,y. Thus, for all S, we have

(TS-ST)x,y = TSx,y-Tx,S*y
= lim(TiSx,y-Tix,S*y)
= lim(TiS-STi)x,y
= lim(TiS-TiS)x,y
= 0

Hence, TS-ST=0, so that T. We conclude that is closed in the weak operator topology.

Title commutant is a weak operator closed subalgebra
Canonical name CommutantIsAWeakOperatorClosedSubalgebra
Date of creation 2013-03-22 18:39:32
Last modified on 2013-03-22 18:39:32
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46L10