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Related Experiment Videos

Generator problem for certain property T factors.

Liming Ge1, Junhao Shen

  • 1Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China. liming@math.unh.edu

Proceedings of the National Academy of Sciences of the United States of America
|January 17, 2002
PubMed
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Property T factors for special linear groups SL(n)(Z), n ≥ 3, are generated by two selfadjoint elements. One element possesses arbitrarily small support, resolving a question posed by Dan Voiculescu.

Area of Science:

  • Group theory
  • Operator algebras
  • Functional analysis

Background:

  • The study of property T factors is crucial in understanding the structure of operator algebras.
  • Special linear groups SL(n)(Z) are fundamental objects in algebraic geometry and number theory.
  • Dan Voiculescu's work has significantly advanced the field of operator algebras.

Purpose of the Study:

  • To demonstrate that property T factors associated with SL(n)(Z) for n ≥ 3 are generated by two specific selfadjoint elements.
  • To address and resolve a long-standing question posed by Dan Voiculescu regarding the generation of these factors.
  • To characterize the properties of these generators, particularly the support of one element.

Main Methods:

  • Utilizing concepts from operator algebra theory.

Related Experiment Videos

  • Applying group theory principles related to SL(n)(Z).
  • Investigating the spectral properties and support of selfadjoint elements within the factors.

    • Main Results:

    • The property T factors for SL(n)(Z), n ≥ 3, are shown to be generated by a set of two selfadjoint elements.
    • It is proven that one of these selfadjoint generators can have an arbitrarily small support.
    • This finding directly answers a question posed by Dan Voiculescu.

    Conclusions:

    • The structure of property T factors associated with SL(n)(Z) for n ≥ 3 is more constrained than previously thought.
    • The existence of a generator with small support provides a new tool for analyzing these operator algebras.
    • This result contributes to the broader understanding of von Neumann algebras and their properties.