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Quasi-invariant states

Luigi Accardi, Ameur Dhahri

Abstract

We develop the theory of quasi--invariant (resp. strongly quasi--invariant) states under the action of a group $G$ of normal $*$--automorphisms of a $*$--algebra (or von Neumann alegbra) $\mathcal{A}$. We prove that these states are naturally associated to left--$G$--$1$--cocycles. If $G$ is compact, the structure of strongly $G$--quasi--invariant states is determined. For any $G$--strongly quasi--invariant state $\varphi$, we construct a unitary representation associated to the triple $(\mathcal{A},G,\varphi)$. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and hermitean conditional density amplitudes on a countable tensor product of type I factors is strongly quasi--invariant with respect to the natural action of the group $\mathcal{S}_{\infty}$ of local permutations and we give the explicit form of the associated cocycle. This provides a family of non--trivial examples of strongly quasi--invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.

Quasi-invariant states

Abstract

We develop the theory of quasi--invariant (resp. strongly quasi--invariant) states under the action of a group of normal --automorphisms of a --algebra (or von Neumann alegbra) . We prove that these states are naturally associated to left------cocycles. If is compact, the structure of strongly --quasi--invariant states is determined. For any --strongly quasi--invariant state , we construct a unitary representation associated to the triple . We prove, under some conditions, that any quantum Markov chain with commuting, invertible and hermitean conditional density amplitudes on a countable tensor product of type I factors is strongly quasi--invariant with respect to the natural action of the group of local permutations and we give the explicit form of the associated cocycle. This provides a family of non--trivial examples of strongly quasi--invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.
Paper Structure (11 sections, 21 theorems, 206 equations)

This paper contains 11 sections, 21 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Quasi--invariant states under a group of $*$-automorphisms
  3. The case $\mathcal{A}=\mathcal{B}(\mathcal{H})$
  4. Quasi--invariant states with trivial cocycles
  5. Examples of quasi--invariant states under permutations
  6. Product states
  7. Markov Chains with commuting conditional density amplitudes
  8. Unitary representations associated to strongly quasi--invariant states
  9. Structure of strongly quasi--invariant states
  10. The case of compact $G$
  11. Inductive limits of compact groups

Key Result

Proposition 1

Suppose that, for any $g\in G$, there exists a $x_{g}\in\mathcal{A}$ such that Then the map $g\in G\mapsto x_{g}\in\mathcal{A}$ is a normalized multiplicative left $G$--$1$--cocycle, i.e. it satisfies the identities In particular each $x_{g}$ is invertible and its inverse is

Theorems & Definitions (28)

  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Definition 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Definition 2
  • Lemma 4
  • Lemma 5
  • ...and 18 more