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Topological and bornological characterisations of ideals in von Neumann algebras: I
Authors:Graeme West
Affiliation:(1) Dept Math and Computer Science, Kent State University, 44242 Kent, OH, USA
Abstract:
Suppose
$$mathcal{M}$$
is a von Neumann algebra on a Hilbert space
$$mathcal{H}$$
and
$$mathcal{I}$$
is any ideal in
$$mathcal{M}$$
. We determine a topology
$$t(mathcal{I})$$
on
$$mathcal{H}$$
, for which the members of
$$mathcal{M}$$
that are
$$t(mathcal{I})$$
to norm continuous are exactly those in
$$mathcal{I}$$
; and a bornology
$$b(mathcal{I})$$
on
$$mathcal{H}$$
such that the elements of
$$mathcal{M}$$
which map the unit ball to an element of
$$b(mathcal{I})$$
, equivalently those members of
$$mathcal{M}$$
that are norm to
$$b(mathcal{I})$$
bounded, are exactly those in
$$mathcal{I}$$
. This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.
Keywords:primary 46L10  secondary 46A17  47D50
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