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Gap-labelling conjecture with nonzero magnetic field
Given a constant magnetic field on Euclidean space determined
by a skew-symmetric matrix , and a -invariant probability measure on the disorder set which is
by hypothesis a Cantor set, where the action is assumed to be minimal, the
corresponding Integrated Density of States of any self-adjoint operator
affiliated to the twisted crossed product algebra , where is the multiplier on associated
to , takes on values on spectral gaps in the magnetic gap-labelling
group. The magnetic frequency group is defined as an explicit countable
subgroup of involving Pfaffians of and its sub-matrices.
We conjecture that the magnetic gap labelling group is a subgroup of the
magnetic frequency group. We give evidence for the validity of our conjecture
in 2D, 3D, the Jordan block diagonal case and the periodic case in all
dimensions.Comment: 43 pages. Exposition improve
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