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Operator Paving


Operator paving over a maximal abelian subalgebra A subset= M subset= B(H) partitions the identity into mutually orthogonal projections p_1,...,p_r in A to make the pinching

 C_P(x)=sum_(i=1)^rp_ixp_i

close to a diagonal element a in A. Norm paving bounds ||C_P(x)-a|| directly. Strong-operator paving, for self-adjoint x, requires that for every finite F subset H and delta>0 a projection q in M can be chosen with ||(1-q)xi||<delta for xi in F and ||a||<=||x|| such that

 ||q(C_P(x)-a)q||<=epsilon||x||.

The partition count is chosen before F and delta.


See also

Maximal Abelian Subalgebra, Operator Norm, Strong Operator Topology

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References

Popa, S. and Vaes, S. "Paving over Arbitrary MASAs in von Neumann Algebras." Anal. PDE 8, 1001-1023, 2015. https://doi.org/10.2140/apde.2015.8.1001.

Cite this as:

Weisstein, Eric W. "Operator Paving." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OperatorPaving.html

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