Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 145-169.

Distance between unitary orbits of self-adjoint elements in $C^*$-algebras of generalized tracial rank one

Authors: Ruofei Wang (1), Sihan Wei (2)
Author institution: (1) College of Data Science, Jiaxing University, Jiaxing, 314-100, China
(2) Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China

Summary: The article investigates the relationship between the distance of unitary orbits and spectral distance in $C^*$-algebras of generalized tracial rank one. Building on the classical result of H. Weyl, which establishes that for Hermitian matrices $x$ and $y$, the distance between their unitary orbits $\text{dist}(U(x),$ $ U(y))$ coincides with their spectral distance $\delta(x, y)$, subsequent extensions have been made: R. Wang proved the equality for simple $C^*$-algebras of tracial rank one, and C.W. Cheong for one-dimensional noncommutative finite CW complexes. In this work, we further extend the result to simple $C^*$-algebras of generalized tracial rank one, showing that for any self-adjoint elements $x, y$, the equality $ \mathrm{dist}(U(x), U(y)) = D_\mathrm c(x, y) $ continues to hold.

DOI: http://dx.doi.org/10.7900/jot.2025jan26.2502
Keywords: unitary orbits, spectral distance, generalized tracial rank

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