Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-commutative measure theory: Henkin and analytic functionals on
  $\mathrm{C}^*$-algebras   Henkin functionals on non-commutative $\mathrm{C}^*$-algebras have recently
emerged as a pivotal link between operator theory and complex function theory
in several variables. Our aim in this paper is characterize these functionals
through a notion of absolute continuity, inspired by a seminal theorem of Cole
and Range. To do this, we recast the problem as a question in non-commutative
measure theory. We develop a Glicksberg--K\"onig--Seever decomposition of the
dual space of a $\mathrm{C}^*$-algebra into an absolutely continuous part and a
singular part, relative to a fixed convex subset of states. Leveraging this
tool, we show that Henkin functionals are absolutely continuous with respect to
the so-called analytic functionals if and only if a certain compatibility
condition is satisfied by the ambient weak-$*$ topology. In contrast with the
classical setting, the issue of stability under absolute continuity is not
automatic in this non-commutative framework, and we illustrate its key role in
sharpening our description of Henkin functionals. Our machinery yields new
insight when specialized to the multiplier algebras of the Drury--Arveson space
and of the Dirichlet space, and to Popescu's noncommutative disc algebra. As
another application, we make a contribution to the theory of non-commutative
peak and interpolation sets.

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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph