Groupoid Characterization of Partial Algebras on Sobolev Spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915417238274048 |
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| author | Okeke, N. O. Egwe, M. E. |
| author_facet | Okeke, N. O. Egwe, M. E. |
| contents | The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(Ω), Γ, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(Ω)$, delineated by weak derivatives as subspaces of $L^p$-spaces is shown to contain the partial algebra $(L^p(Ω), Γ, \cdot)$ generalized by the partial action of the smooth algebra $\mathscr{K}(Ω)$ by convolution on the Banach spaces $L^p(Ω)$. We characterised the Sobolev space $W^{k,p}(Ω)$, invariant under $\mathscr{K}(Ω)$ partial action, using Lie groupoid framework, and study the partial algebra as defining the partial dynamical systems on the $L^p$-space associated with the weak differential operators. The locally convex partial $^*$-algebra $(L^p(Ω), Γ, \cdot,^*)$ defines the stable local flows coinciding with local bisections of the Lie groupoid. The unitary representation of resulting Lie groupoid $\mathscr{W} \rightrightarrows W^{k,p}(Ω)$ on the associated Hilbert bundle demonstrates the simplification achieved by the characterisation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18436 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groupoid Characterization of Partial Algebras on Sobolev Spaces Okeke, N. O. Egwe, M. E. Functional Analysis Operator Algebras 22A22, 58H05, 46E35, 47L60 The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(Ω), Γ, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(Ω)$, delineated by weak derivatives as subspaces of $L^p$-spaces is shown to contain the partial algebra $(L^p(Ω), Γ, \cdot)$ generalized by the partial action of the smooth algebra $\mathscr{K}(Ω)$ by convolution on the Banach spaces $L^p(Ω)$. We characterised the Sobolev space $W^{k,p}(Ω)$, invariant under $\mathscr{K}(Ω)$ partial action, using Lie groupoid framework, and study the partial algebra as defining the partial dynamical systems on the $L^p$-space associated with the weak differential operators. The locally convex partial $^*$-algebra $(L^p(Ω), Γ, \cdot,^*)$ defines the stable local flows coinciding with local bisections of the Lie groupoid. The unitary representation of resulting Lie groupoid $\mathscr{W} \rightrightarrows W^{k,p}(Ω)$ on the associated Hilbert bundle demonstrates the simplification achieved by the characterisation. |
| title | Groupoid Characterization of Partial Algebras on Sobolev Spaces |
| topic | Functional Analysis Operator Algebras 22A22, 58H05, 46E35, 47L60 |
| url | https://arxiv.org/abs/2405.18436 |