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A quantum remark on biholomorphic mappings on the unit ball

Sebastian Schleißinger

Abstract

In this note we regard non-commutative probability theory with operator-valued expectation. We show that the moment generating functions of distributions coming from monotone increment processes of unitary random variables yield biholomorphic mappings on certain higher dimensional unit balls.

A quantum remark on biholomorphic mappings on the unit ball

Abstract

In this note we regard non-commutative probability theory with operator-valued expectation. We show that the moment generating functions of distributions coming from monotone increment processes of unitary random variables yield biholomorphic mappings on certain higher dimensional unit balls.

Paper Structure

This paper contains 5 sections, 6 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Normalized Loewner chains on unit balls in $\mathbb{C}^n$
  3. Quantum probability
  4. Loewner chains from monotone increment processes
  5. Multivariate processes

Key Result

Theorem 2.5

The class $\mathcal{M}$ is compact with respect to locally uniform convergence.

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5: Theorem 6.1.39 in GK03
  • Theorem 2.6
  • proof
  • Example 3.1
  • Lemma 3.2
  • proof
  • ...and 17 more