Lebesgue decomposition of noncommutative measures
Abstract
The RieszMarkov theorem identifies any positive, finite, and regular Borel measure on the complex unit circle with a positive linear functional on the continuous functions. By the Weierstrass approximation theorem, the continuous functions are obtained as the norm closure of the Disk Algebra and its conjugates. Here, the Disk Algebra can be viewed as the unital normclosed operator algebra of the shift operator on the Hardy Space, $H^2$ of the disk. Replacing squaresummable Taylor series indexed by the nonnegative integers, i.e. $H^2$ of the disk, with squaresummable power series indexed by the free (universal) monoid on $d$ generators, we show that the concepts of absolutely continuity and singularity of measures, Lebesgue Decomposition and related results have faithful extensions to the setting of `noncommutative measures' defined as positive linear functionals on a noncommutative multivariable `Disk Algebra' and its conjugates.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.09965
 Bibcode:
 2019arXiv191009965J
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras
 EPrint:
 arXiv admin note: text overlap with arXiv:1907.09590