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Related Experiment Videos

Traces, ideals, and arithmetic means.

Victor Kaftal1, Gary Weiss

  • 1Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA. kaftal@math.uc.edu

Proceedings of the National Academy of Sciences of the United States of America
|May 29, 2002
PubMed
Summary

This study explores operator ideals using arithmetic means (am), revealing properties like am-stability and am-closure. It establishes that the linear codimension of commutator spaces is 0, 1, or infinity, with applications to elementary operators.

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Area of Science:

  • Operator Theory
  • Functional Analysis
  • Abstract Algebra

Background:

  • Recent work characterized commutator ideals using arithmetic means.
  • Operator ideals exhibit various properties related to arithmetic mean operations.

Purpose of the Study:

  • To investigate operator ideals with properties such as arithmetic mean (am) stability, am-closure, am-openness, soft-edgedness, and soft-complementation.
  • To apply these properties to determine the linear codimension of commutator spaces.
  • To identify ideals supporting unique nonsingular traces and explore applications to elementary operators.

Main Methods:

  • Characterization of operator ideals based on arithmetic mean properties.
  • Analysis of linear codimension and nonsingular traces.

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  • Investigation of arithmetic mean operations on sums and cancellations of ideals.
  • Construction of counterexamples for cancellation properties.

    • Main Results:

    • Many operator ideals possess am-stability, am-closure, am-openness, soft-edgedness, and soft-complementation.
    • The linear codimension of commutator spaces for considered ideals is proven to be 0, 1, or infinity.
    • The largest ideal with a unique nonsingular trace is identified as an intersection of Lorentz ideals.
    • Properties of arithmetic mean operations on sums of ideals are established, including am-closure.
    • Cancellation properties for arithmetic means are analyzed, with conditions for first and second-order cancellations identified.
    • A counterexample demonstrating the failure of second-order cancellation is constructed, resolving an open question.

    Conclusions:

    • The study provides a comprehensive analysis of operator ideals through the lens of arithmetic means.
    • The findings offer insights into the structure of commutator spaces and the existence of unique traces.
    • Established cancellation properties and counterexamples contribute to a deeper understanding of ideal theory in operator algebras.