Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dilations for operator-valued quantum measures   This paper concerns the dilations of Banach space operator-valued quantum
measures. While the recently developed general dilation theory can lead to a
projection (idempotent) valued dilation for any quantum measure over the
projection lattice for a von Neumann algebra that dose not contain type $I_{2}$
direct summand, such a dilation does not necessarily guarantee the preservation
of countable additivity of the quantum measure. So it remain an open question
whether every countably additive $B(X)$-valued quantum measure can be dilated
to a countably additive projection-valued measure.The main purpose of this
paper is to prove that such a dilation can be constructed if one of the
following two conditions is satisfied: (i) the underling Banach space $X =
\ell_{p}$ $(1\leq p < 2$) or it has Schur property, (ii) the quantum measure
has bounded $p$-variation for some $ 1\leq p < \infty $. All of these were
achieved by establishing a non-commutative version of a minimal dilation theory
on the so-called elementary dilation space equipping with an appropriate
dilation norm. In particular, the newly introduced $p$-variation norm on the
elementary dilation space allows us to prove that every operator-valued quantum
measure with bounded $p$-variation has a projection-valued quantum measure
dilation that preserves the boundedness of the $p$-variation.

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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys