Abstract
The present paper is devoted to the study of the dimension of the space of all derivations on homogeneous commutative regular algebras. We shall show that if \(\left( \Omega , \Sigma , \mu \right)\) is a Maharam homogeneous measure space with a finite countable-additive measure \(\mu\) and \(\mathcal {A}\) is a homogeneous regular unital subalgebra in \(S(\Omega )\) with the homogeneous Boolean algebra of idempotents \(\nabla (\mathcal {A}),\) then \(\dim \textrm{Der}(\mathcal {A})=\tau (\nabla (\mathcal {A}))^{\textrm{trdeg}(\mathcal {A})},\) where \(\tau (\nabla (\mathcal {A}))\) is the weight of the Boolean algebra \(\nabla (\mathcal {A})\) and \(\textrm{trdeg}(\mathcal {A})\) is the transcendence degree of \(\mathcal {A}\).
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https://mathoverflow.net/questions/49551/dimension-of-infinite-product-of-vector-spaces.
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Acknowledgements
We are indebted to the referees for very valuable suggestions and remarks, which helped us to significantly improve the exposition.
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The second author was partially supported by the Russian Ministry of Education and Science, agreement no. 075-02-2023-914.
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With deep respect, we dedicate the article to the 70th anniversary of Professor Anatoly Georgievich Kusraev.
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Ayupov, S., Kudaybergenov, K. & Karimov, K. ON DIMENSION OF THE SPACE OF DERIVATIONS ON COMMUTATIVE REGULAR ALGEBRAS. J Math Sci 271, 694–699 (2023). https://doi.org/10.1007/s10958-023-06577-w
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DOI: https://doi.org/10.1007/s10958-023-06577-w
Keywords
- Homogeneous commutative regular algebra
- Maharam measure space
- Algebra of measurable functions
- Derivation