Let H be a Hilbert space and <mi mathvariant="script">A a commutative norm-closed unital

Lorena Beard 2022-07-10 Answered
Let H be a Hilbert space and A a commutative norm-closed unital -subalgebra of B ( H ). Let M be the weak operator closure of A.
Question: For given a projection P M, is the following true?
P = inf { A A : P A 1 }
It seems that the infimum must exist and is a projection, but I am not able to show that the resulting projection cannot be strictly bigger than P. Also, if the above is true, what happens if A is non-commutative?
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Answers (1)

Oliver Shepherd
Answered 2022-07-11 Author has 24 answers
Take M = L [ 0 , 1 ] and A = C [ 0 , 1 ].
Let Q [ 0 , 1 ] = { r n } n = 1 be an enumeration of rationals in [ 0 , 1 ], and define
E := n = 1 ( r n ϵ 2 n , r n + ϵ 2 n ) [ 0 , 1 ]
for some small ϵ > 0 so that m ( E ) is strictly less than 1. Clearly, the only continuous function f : [ 0 , 1 ] [ 0 , 1 ] satisfying 1 E f 1 is f 1, but the projection 1 E is not equal to 1.
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