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Ábel Komálovics (BME): A one parameter extension of the Bures and Hellinger distances, and trace characterisations

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Tuesday, 8. November 2022, 11:00 - 12:00
We study a one parameter extension of the Bures and Hellinger distances.
It is the square root of the trace of the difference of the arithmetic
mean and a sort of geometric mean. We characterize the parameters for
which the extension is a true metric on the pure states of a Hilbert
space. After this, we examine whether the extension is well-defined in
the far more general C*-algebraic context. We present two statements
concerning the order between the arithmetic mean and the variants of the
geometric mean appearing in the extension in question. One of them
characterises the central elements of a C*-algebra, the other one
characterises the traciality of positive linear functionals. Our last
statement characterises the tracial self-adjoint linear functionals of a
von Neumann algebra which, in the finite dimensional case, can also be
viewed as a characterisation of the determinant.

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