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单词 CommutantIsAWeakOperatorClosedSubalgebra
释义

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.

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更新时间:2025/5/4 5:26:26