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Isono Lab
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The lab explores how von Neumann algebras built from discrete groups relate to ergodic theory.
AboutAbout the lab
Studying operator algebras that arise from group actions
The Isono Lab studies the relationship between von Neumann algebras and ergodic theory, in the field of operator algebras.
An operator algebra is a subalgebra of the algebra of all bounded linear operators on a Hilbert space. The concept first appeared when von Neumann gave a mathematical formulation of quantum mechanics.
The lab mainly investigates von Neumann algebras constructed from discrete groups, from the viewpoint of deformation/rigidity theory.
Key terms
- Operator algebra
- An algebraic structure formed by collecting linear transformations (operators) that act on an infinite-dimensional space (a Hilbert space).
- Ergodic theory
- A branch of mathematics that studies the average long-term behavior of a system when a transformation is applied repeatedly.
- Rigidity theorem
- A theorem showing that objects that can be regarded as the same in a weak sense are in fact the same in a strong sense as well.
ResearchResearch themes
The lab studies the relationship between von Neumann algebras and ergodic theory. Its work mainly uses deformation/rigidity theory, which deals with von Neumann algebras constructed from actions of discrete groups on measure spaces.
*Illustrative imageDeformation/rigidity theory
For example, S. Popa's rigidity theorem shows, for Bernoulli shift actions of discrete groups with Kazhdan's property (T), a phenomenon that does not occur for the integers.
*Illustrative imageStructure and rigidity of von Neumann algebras
The lab has also succeeded in developing the theory by using Tomita–Takesaki theory, and has proved a type of rigidity theorem that can be obtained only by way of Tomita–Takesaki theory.
MembersFaculty
- Associate ProfessorYusuke Isono
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Bernoullishift” Andreas Mandl / CC BY-SA 3.0 (cropped)
- Deformation/rigidity theory: “Ergodic mixing of putty ball after repeated Smale horseshoe map” Yaroslav I. Sobolev and Kamil R. Karimullin / CC BY-SA 3.0 (cropped)
- Structure and rigidity of von Neumann algebras: “Hyperbolic tiling 9-2 9” Double sharp (talk). Created using Tyler and Paint. / Public domain (cropped)
- Related lab (Ozawa Lab): “Cayley graph of Alternating Group A5” Kathryn Cramer / CC BY-SA 4.0 (cropped)
- Related lab (Sehnem Lab): “Hyperbolic tiling 7 7-2” Double sharp (talk), based on this picture by Jim McNeill, but with the remaining overlapping heptagons left as outlines instead of deleted. A green background to the tiling has also been added. Created using Tyler and Paint. / Public domain (cropped)
- Related lab (Applied Mathematics Lab (Kazuhisa Makino)): “B functies 2var versie02” Bob.v.R / CC BY-SA 4.0 (cropped)
- Related lab (Applied Mathematics Division (Kobayashi) Lab): “Graph-matroid” Didia / CC BY-SA 4.0 (cropped)
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