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Non-commutative Lebesgue decomposition of non-commutative measures

Fouad Naderi

Abstract

A positive non-commutative (NC) measure is a positive linear functional on the free disk operator system which is generated by a $d$-tuple of non-commuting isometries. By introducing the hybrid forms, their Cauchy transforms, and techniques from NC reproducing kernel Hilbert spaces (RKHS), we construct a natural Lebesgue decomposition for any positive NC measure against any other such measure. Our work extends the Jury-Martin decomposition, which originally decomposes positive NC measures against the standard NC Lebesgue measure. In fact, we give a more generalized definition of absolute continuity and singularity, which reduces to their definition when the splitting measure is the standard NC Lebesgue measure. This generalized definition makes it possible to extend Jury-Martin theory for any splitting NC measure, and it recovers their decomposition when the splitting NC measure is the Lebesgue one. Our work implies a Lebesgue decomposition for representations of the Cuntz-Toeplitz C*-algebra. Furthermore, our RKHS method gives a new proof of the classical Lebesgue decomposition when applied to the classical one dimensional setting, i.e., $d=1$.

Non-commutative Lebesgue decomposition of non-commutative measures

Abstract

A positive non-commutative (NC) measure is a positive linear functional on the free disk operator system which is generated by a -tuple of non-commuting isometries. By introducing the hybrid forms, their Cauchy transforms, and techniques from NC reproducing kernel Hilbert spaces (RKHS), we construct a natural Lebesgue decomposition for any positive NC measure against any other such measure. Our work extends the Jury-Martin decomposition, which originally decomposes positive NC measures against the standard NC Lebesgue measure. In fact, we give a more generalized definition of absolute continuity and singularity, which reduces to their definition when the splitting measure is the standard NC Lebesgue measure. This generalized definition makes it possible to extend Jury-Martin theory for any splitting NC measure, and it recovers their decomposition when the splitting NC measure is the Lebesgue one. Our work implies a Lebesgue decomposition for representations of the Cuntz-Toeplitz C*-algebra. Furthermore, our RKHS method gives a new proof of the classical Lebesgue decomposition when applied to the classical one dimensional setting, i.e., .
Paper Structure (5 sections, 35 theorems, 168 equations, 1 figure)

This paper contains 5 sections, 35 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Lebesgue reproducing kernel Hilbert space decomposition with respect to a non-Cuntz measure
  4. The Lebesgue form decomposition with respect to a non-Cuntz measure
  5. Lebesgue decomposition with respect to a Cuntz measure

Key Result

Theorem 1

For any non-Cuntz measure $\tau$, the left ideal of zero length elements, $N_{\tau}$, is zero.

Figures (1)

  • Figure 1: relationship among various theories

Theorems & Definitions (95)

  • Definition 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • Remark 3
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 85 more