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Ergodic decompositions of Dirichlet forms under order isomorphisms.

Lorenzo Dello Schiavo1, Melchior Wirth1

  • 1Institute of Science and Technology Austria, Klosterneuburg, Austria.

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|January 4, 2023
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Ergodic decompositions of quasi-regular Dirichlet spaces are unique. Unitary isomorphisms intertwining these spaces decompose over their ergodic structures, ensuring a unique mapping between indexing spaces.

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Direct integralDirichlet formsErgodic decompositionIntertwiningOrder isomorphism

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Area of Science:

  • Functional Analysis
  • Operator Algebras
  • Stochastic Processes

Background:

  • Dirichlet spaces are fundamental in analysis and probability.
  • Ergodic theory studies the long-term average behavior of dynamical systems.
  • Understanding decompositions is key to classifying complex mathematical structures.

Purpose of the Study:

  • To investigate the uniqueness and structure of ergodic decompositions in quasi-regular Dirichlet spaces.
  • To analyze the properties of unitary order isomorphisms that intertwine such spaces.
  • To establish the relationship between these isomorphisms and the ergodic decompositions.

Main Methods:

  • Utilizing the theory of Dirichlet spaces.
  • Applying concepts from ergodic theory and functional analysis.
  • Developing techniques for decomposing unitary order isomorphisms.

Main Results:

  • The ergodic decomposition of a quasi-regular Dirichlet space is unique up to isomorphism.
  • Every unitary order isomorphism between two quasi-regular Dirichlet spaces can be decomposed over their respective ergodic decompositions.
  • This decomposition is unique up to conjugation by an isomorphism of the indexing spaces.

Conclusions:

  • The study provides a rigorous framework for understanding ergodic decompositions in Dirichlet spaces.
  • The results offer insights into the structure of intertwining operators between these spaces.
  • This work contributes to the classification and analysis of complex mathematical objects.