Notes about math, research, and more.
:tags: research-rufus, research, exotic-crossed-products

LaTeX files

  1. appendix.tex
    • part of consolidated-notes.tex
  2. auslander.tex
    • What You would expect, just notes from the Auslander textbook
    • Was created last week (Nov 24)
  3. big-picture.tex
    • Nice; what I printed out to go over with Rufus.
  4. consolidated-notes.tex
    • Pretty nicely done. Should be useful
  5. exotic-crossed-products.tex
  6. fibers.tex
    • Merged into consolidatednotes
  7. group-extension-work.tex
    • Important nice work here
  8. gwep.tex
    • Some nice nots. No research that I can see – just notes
  9. may-13.tex
  10. may-20.tex
  11. weakly-almost-periodic.tex
  12. paper/paper.tex

Notes

  1. The /paper/paper.tex file is split into a main paper.tex
    • ./lemma-1/main-1.tex
    • ./lemma-1/mu.tex
    • ./lemma-2/main-2.tex

Fixes

  1. A lemma that beings in main-1.tex is closed in either mu.tex (more likely) or main-2.tex.

Todo

  1. Fill in main theorem

Paper

Goal

Goal of the paper is to find \(C^\ast\text{-algebras}\) \(A\) and discrete group \(G\) which acts on \(A\) such that \(A \rtimesimes_{\ex} G = A\rtimes_{\inj} G\).

Methods

For a nuclear \(G-C^*{-algebra}\) \(A\) and a discrete group \(G\) acting on \(A\), the following commutative diagram exists:

\begin{tikzcd} (\ell^\infty(G)\otimes A)\rtimes_{\max} G \arrow[r, "(\mathlarger{*})"] \arrow[d, hookleftarrow, "\parbox{2cm}{\footnotesize Kranz-Siebenand}", "\parbox{1.2cm}{\footnotesize always\\ injective}"'] & (\ell^\infty(G,A^{**}))\rtimes_{\max} G \\ A\rtimes_{\ex} G \arrow[r, ->>, "\psi"'] & A\rtimes_{\inj} G \arrow[u, "\parbox{2cm}{\footnotesize $A$ nuclear gives\\ existance}"'] \end{tikzcd}