Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi-Cosimplicial Hilbert Spaces with Isometric Coface Operators

D. Gwion Evans, Rolf Gohm, Claus Köstler

Abstract

Semi-cosimplicial objects in the category of Hilbert spaces with isometries which are motivated by non-commutative probability theory, in particular by the distributional symmetry of spreadability, are introduced and systematically developed in various directions: partial shifts, cohomology, a related graph, decomposition into labeled subspaces, representation theory of the infinite symmetric and braid groups, extension problems and a toy version of spreadability.

Semi-Cosimplicial Hilbert Spaces with Isometric Coface Operators

Abstract

Semi-cosimplicial objects in the category of Hilbert spaces with isometries which are motivated by non-commutative probability theory, in particular by the distributional symmetry of spreadability, are introduced and systematically developed in various directions: partial shifts, cohomology, a related graph, decomposition into labeled subspaces, representation theory of the infinite symmetric and braid groups, extension problems and a toy version of spreadability.

Paper Structure

This paper contains 9 sections, 26 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. SCHs, Partial Shifts, Saturation and a Toy de Finetti Theorem
  3. Cohomology
  4. Hessenberg Factorizations and Braided SCHs
  5. The graph of $\Delta_{S}$
  6. Labeled Subspaces
  7. Normal SCHs and tame representations of the infinite symmetric group $\mathbb{S}_\infty$
  8. SCSs and Extensions
  9. Spreadability wrt an inner product

Key Result

Proposition 2.6

Any sequence $(\alpha_n)_{n \in \mathbb{N}_0}$ of partial shifts is associated to the saturated SCH $({\mathcal{H}}\xspace_k, \delta_i)_{k,i}$ given by ${\mathcal{H}}\xspace_{k} := {\mathcal{H}}\xspace^{\alpha_{k+1}}$ for all $k$.

Figures (2)

  • Figure 1: The skeleton of $\Lambda_S$ (only the rank 0, 1 and 2 components are shown). Dashed lines are to indicate the level of each vertex only; they are not part of the graph. The label on each vertex is its characterstic function written as a finite binary sequence. The label of a root is in bold.
  • Figure 2: A non-normal, saturated SCS.

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Definition 2.8
  • Definition 2.10
  • ...and 59 more