arXiv subjects

N. O. Okeke

Publications and source records attributed to N. O. Okeke.

5 recordsLinked to original sources

arXiv2025

Partial Dynamical Systems of $L^p$-Spaces and their Stability Spaces

Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(Ω)$ spaces defined by the action of the smooth algebra $\mathscr{K}(Ω)$ through its nets. Slice analysis is then employed to show that the Sobolev spaces $W^{k,p}(Ω)$ are the stable states or space of these partial dynamical systems as limit spaces of the convolution actions of the smooth algebra $K(Ω)$ on the Banach spaces $L^p(Ω)$. Thus, the Sobolev spaces $W^{k,p}(Ω)$ are closed subspaces of the $Lp(Ω)$-spaces under convolution product and weak derivatives, with the weak derivative operators acting as equivariant maps of the slice spaces.

math.FA↗

arXiv2025

Groupoid Characterization of Partial Algebras on Sobolev Spaces

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.

math.FA↗

arXiv2021

Groupoid approach to the dynamical system of commutative von Neumann Algebras

The automorphism group $Aut(X,μ)$ of a compact, complete metric space $X$ with a Radon measure $μ$ is a subgroup of $\mathcal{U}(L^2(X,μ))$-the unitary group of operators on $L^2(X,μ)$. The $Aut(X,μ)$-action on the generalized space $\mathcal{M}(X)$ is a proper action. Hence, there exists a slice at each point of the generalized space $\mathcal{M}(X)$. Measure Groupoid (virtual group) is subsequently employed to analyze the resulting dynamical system as that of the ergodic action of the commutative algebra (a lattice) $C(X)$ on the generalized space $\mathcal{M}(X)$ which is represented on a commutative von Neumann algebra.

math.DS↗

arXiv2021

Groupoid Characterization of Locally Convex Partial $^*$-Algebras

Given a locally convex space $(\mathcal{A},τ)$ with a Hausdorff locally convex topology $τ$ such that the following maps are continuous; $u \mapsto u^*$ for all $u \in \mathcal{A}$, $x \mapsto x\cdot y$ and $x \mapsto z\cdot x$ for every left and right multipliers of $\mathcal{A}$. In this paper we re-characterized the locally convex partial $*$-algebra $(\mathcal{A}, Γ,\cdot,*,τ)$ arising from these continuous maps in terms of convolution algebra of a Lie groupoid $Γ\rightrightarrows \mathcal{A}$. This is advantageous because the pathologies of the underlying spaces owing to their quantum mechanical nature are easily resolved in groupoid terms.

math.OA↗

arXiv2021

Cohomogeneity One Groupoid Analysis of the Dynamical System of Rings of Continuous Functions

Using the group $G(1)$ of invertible elements and the maximal ideals $\mathfrak{m}_x$ of the commutative algebra $C(X)$ of real-valued functions on a compact regular space $X$, we define a Borel action of the algebra on the measure space $(X,μ)$ with $μ$ a Radon measure. The zero sets $Z(X)$ of the algebra $C(X)$ is used to study the ergodicity of the $G(1)$-action via its action on the maximal ideals $\mathfrak{m}_x$ which defines an action groupoid $\mathcal{G} = \mathfrak{m}_x \ltimes G(1)$ trivialized on $X$. The resulting measure groupoid $(\mathcal{G},\mathcal{C})$ is used to define a proper action on the generalized space $\mathcal{M}(X)$. The existence of slice at each point of $\mathcal{M}(X)$ present it as a cohomogeneity-one $\mathcal{G}$-space. The dynamical system of the algebra $C(X)$ is defined by the action of the measure groupoid $(\mathcal{G},\mathcal{C}) \times \mathcal{M}(X) \to \mathcal{M}(X)$.

math.FA↗