Matrix-valued measures and integration on foundation semigroups

Document Type : Original Article

Author

Kyrgyz-Turkish Manas University, Bishkek, Kyrgyzstan

Abstract

This paper develops a comprehensive theory of matrix-valued measures and integration on foundation topological semigroups, extending the well-established framework for topological groups. We establish fundamental results, including polar decomposition, duality theory, and convolution algebras for $M_n$-valued measures. The paper characterizes several important classes of measures ($\Mel$, $\Mer$, $\Mal$, $\Mar$) and analyzes their structural properties, showing that $\Ma$ forms an L-ideal and $\Me$ is an L-subalgebra. Finally, we develop a theory of quasi-invariant matrix-valued measures, proving existence results and characterizing the measure algebra $\Me$ in terms of equi-quasi-invariant measures. Our work provides a unified framework for non-commutative harmonic analysis on semigroups, generalizing both the scalar theory for foundation semigroups and the matrix-valued theory for groups.

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References

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