Lebesgue decomposition of non-commutative measures

TL;DR

This paper extends classical measure theory concepts like Lebesgue decomposition to non-commutative measures defined on operator algebras associated with free monoids, broadening the scope of measure theory in non-commutative analysis.

Contribution

It introduces a non-commutative Lebesgue decomposition theory for positive linear functionals on multi-variable Disk Algebras, generalizing classical measure concepts.

Findings

Established non-commutative analogs of absolute continuity and singularity.

Extended Lebesgue decomposition to non-commutative measures.

Connected classical measure theory with operator algebra frameworks.

Abstract

The Riesz-Markov 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 norm-closed operator algebra of the shift operator on the Hardy Space, $H^2$ of the disk. Replacing square-summable Taylor series indexed by the non-negative integers, i.e. $H^2$ of the disk, with square-summable 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 `non-commutative measures' defined as positive linear functionals on a non-commutative multi-variable `Disk Algebra' and its conjugates.