Finite games and operator-algebra type
Which finite games only type III algebras can win perfectly, and what play reveals about an algebra’s type.
Two works in two archived versions · latest release
The question
The papersin reading order, with what each settled
-
Finite games that read the type of a von Neumann algebra
Seth Douglas and Nidhal Mghirbi · version 1.0.0 · · 36 pages
A finite two-player game with binary scoring and 243 question pairs has perfect play in type III and none when either ambient algebra is semifinite. Two further finite games characterize type III1 through how every faithful normal state plays them, and every positive rational is forced as a modular ratio by some finite game.
- Partly answered Which Schmidt spectra can be self-tested by pseudo-telepathy games?
- Answered Is there a finite game that type III algebras can win perfectly and semifinite algebras cannot?
- Answered Can finite games characterize type III1?
- Answered Is every positive rational forced as a modular ratio by some finite game?
- Open Can any finite game force an irrational modular ratio? posed here
- Open Can a finite game tell type III1/2 from type III2−1/2 at the worst-state level? posed here
- Open What is the sharp robustness constant of the type-III-only game? posed here
- Open How does an algebra’s type constrain the price of perfection? posed here
-
A finite game that normal hyperfinite strategies approach but never win
Seth Douglas and Nidhal Mghirbi · version 1.0.0 · · 17 pages
An explicit finite game with 0/1 scoring (561 and 157 questions, 3,696 question pairs) whose perfect correlation finite-dimensional strategies approach, while no perfect strategy has a normal marginal on an injective (hyperfinite) von Neumann algebra of either player. Perfect play lives on the Bellaterra critical-KMS factor of type III1/2, and every perfect strategy contains an expected copy of the group algebra of the free product of three groups of order two. Under the hypotheses of Halvorson’s Proposition 2.31, quantum-field-theory strategies come arbitrarily close but never win exactly.