Timeline for fixpoint algebras of a permutation action

5 events

when toggle format	what		by	license	comment
Apr 27, 2015 at 10:51	history	edited	Gabor Szabo	CC BY-SA 3.0	added 436 characters in body
Mar 17, 2015 at 23:32	comment	added	David Handelman		Hmm, it looks like the ordered K$_0$ group of the crossed product and fixed products is given by $H^{X(G)}$ (functions from $X(G)$ to $H$) where $H = K_0 (D)$, $X(G)$ is the set of irreducible characters of $G$, and the unique trace is given by $P:f \mapsto \sum f(\chi)\chi(1)$; this determines the ordering too—nonzero $f$ is positive iff $P(f) \in H^+\setminus \{0\}$. This completely determines which AF algebras appear, up to Morita equivalence. As an afterthought, I should check whether the degrees of the irreducibles (the $\chi(1)$) actually appear in the formula.
Mar 16, 2015 at 21:51	comment	added	Gabor Szabo		Oh I see. Seeing it this way is indeed better than having to appeal to classification.
Mar 16, 2015 at 13:30	comment	added	David Handelman		Because the actions are product type, the fp algebras are automatically AF. In this case, the ordering on $H$, K$_0$ (of the crossed product or fixed point algebra), is easy to describe: $H$ is a split extension, $H \to J$ where $J$ is the value group of the UHF (splitting is a rare phenomenon).
Mar 16, 2015 at 10:56	history	answered	Gabor Szabo	CC BY-SA 3.0