The Classical Smith-Ward Problem
Prior state unknown→disproved
Harris had settled the generalized problem in dimension four; this reaches dimension three
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analysis / Operator algebras
The Classical Smith-Ward Problem
The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\mathrm{span}\{1,q(D),q(K)\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.
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The Smith-Ward theorem realizes the first $k$ essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of $k$, equivalently whether the identity map on a three-dimensional operator system $\mathrm{span}\{1,q(D),q(K)\}$ in the Calkin algebra always lifts. It need not: an explicit three-dimensional hyperrigid operator system has no unital completely positive lift, and its dual is the first three-dimensional operator system that fails to be exact.
Harris had settled the generalized problem in dimension four; this reaches dimension three
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