1,983 research outputs found
Dilations of frames, operator valued measures and bounded linear maps
We will give an outline of the main results in our recent AMS Memoir, and
include some new results, exposition and open problems. In that memoir we
developed a general dilation theory for operator valued measures acting on
Banach spaces where operator-valued measures (or maps) are not necessarily
completely bounded. The main results state that any operator-valued measure,
not necessarily completely bounded, always has a dilation to a
projection-valued measure acting on a Banach space, and every bounded linear
map, again not necessarily completely bounded, on a Banach algebra has a
bounded homomorphism dilation acting on a Banach space. Here the dilation space
often needs to be a Banach space even if the underlying space is a Hilbert
space, and the projections are idempotents that are not necessarily
self-adjoint. These results lead to some new connections between frame theory
and operator algebras, and some of them can be considered as part of the
investigation about "noncommutative" frame theory.Comment: Contemporary Mathematics, 21 pages. arXiv admin note: substantial
text overlap with arXiv:1110.583
A Hilbert C*-module admitting no frames
We show that every infinite-dimensional commutative unital C*-algebra has a
Hilbert C*-module admitting no frames. In particular, this shows that
Kasparov's stabilization theorem for countably generated Hilbert C*-modules can
not be extended to arbitrary Hilbert C*-modules.Comment: Minor change. To appear in Bull. Lond. Math. So
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