What question did this study set out to answer?

To develop a unified theory of the curvature defect functional using bounded self-adjoint operators.

May 3, 20260 citationsOpen Access

The Curvature Defect Functional: A Unified Theory

Key Points

To develop a unified theory of the curvature defect functional using bounded self-adjoint operators.
Constructed a curvature operator Rφ on the L2-gradient module H(2)∇ using a Dirichlet form.
Utilized a rigorous Reilly-type identity to establish operator characteristics.
Applied Kato’s Representation Theorem to derive the Weitzenböck decomposition.
Establishes self-adjoint operators at the level of the Weitzenböck decomposition.
Demonstrates a new approach to understanding Dirichlet forms via curvature operators.

Abstract

We construct a canonical bounded self-adjoint operator Rφ —the curvature operator—on the L2-gradient module H(2)∇ associated with a completely Dirichlet form on a von Neumann algebra. The construction proceeds through a rigorous Reilly-type identity and yields the operator Weitzenböck decomposition L =∇∗H∇H+Rφ at the level of self-adjoint operators via Kato’s Representation Theorem.

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Cite This Study

Jangid Kartik (Fri,) studied this question.

synapsesocial.com/papers/69f6e67c8071d4f1bdfc728d https://doi.org/https://doi.org/10.5281/zenodo.19948810

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