Abstract
As a natural generalization of the Fuglede–Kadison determinant, there is an attempt to extend the concept of the determinant to vector states. In this paper, we present the normalized determinant for a state on a unital C\(^*\)-algebra, and by virtue of the Specht ratio, we show a variant of determinant identity of operator geometric means for the normalized determinant of positive invertible operators. We moreover consider its multi-variable versions. Among others, we show the Karcher mean version and the operator power means one.
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Acknowledgements
The authors would like to express their hearty thanks to the referees for their valuable suggestions and comments for revising the manuscript. The second author is partially supported by JSPS KAKENHI Grant Number JP 19K03542.
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Hiramatsu, S., Seo, Y. Determinant for positive operators and operator geometric means. Anal.Math.Phys. 12, 49 (2022). https://doi.org/10.1007/s13324-022-00663-z
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DOI: https://doi.org/10.1007/s13324-022-00663-z
Keywords
- Operator geometric mean
- Normalized determinant
- Karcher mean
- Positive invertible operator
- Operator power mean