Direct Integral of Locally Hilbert Spaces

• Direct integrals of locally Hilbert spaces are mathematical structures that generalize classical Hilbert space integrals using inductive limit constructions, enabling analysis on locally varying spaces.
• They provide a robust framework for decomposable and diagonalizable operators, leading to locally defined von Neumann algebras essential for spectral theory and quantum analysis.
• The theory integrates projective and inductive limit methods with smooth bundle and measure space constructions, offering versatile applications in representation theory, operator semigroups, and quantum mechanics.

A direct integral of locally Hilbert spaces is a mathematical structure that generalizes the classical theory of direct integrals of Hilbert spaces to the context where the component spaces themselves are inductive limits of Hilbert spaces ("locally Hilbert spaces"). This framework is essential for modeling situations in operator theory, representation theory, and quantum analysis where the underlying spaces carry a graded or locally varying Hilbertian structure and for structuring algebras of operators that act fiberwise or diagonally across such spaces.

1. Foundations and Construction

The classical direct integral constructs a Hilbert space from a measurable field of Hilbert spaces $\{H_p\}_{p \in X}$ over a standard measure space $(X, \Sigma, \mu)$, via

$\int_X^\oplus H_p\,d\mu(p)$

with well-defined $L^2$ vector norms. In the locally Hilbert setting, each $H_p$ is itself an inductive limit of Hilbert spaces: $$H_p = \varinjlim_{\alpha \in \Lambda} H_{\alpha, p}$$ where $\Lambda$ is a directed poset. The measure space itself may be replaced by a "locally standard measure space" $(X, \Sigma, \mu)$, defined as a projective limit of measure spaces $(X_\alpha, \Sigma_\alpha, \mu_\alpha)$ with $X = \bigcup_\alpha X_\alpha$ and

$$\Sigma = \{ E \subset X \mid \forall\,\alpha,\, E \cap X_\alpha \in \Sigma_\alpha \}$$

and $\mu(E) = \lim_\alpha \mu_\alpha(E \cap X_\alpha)$. The direct integral—denoted $$\mathcal{D}_{\text{loc}} = \int_X^\oplus H_p\,d\mu(p)$$—is defined as the inductive limit of the direct integrals $$\int_{X_\alpha}^\oplus H_{\alpha,p}\,d\mu_\alpha(p)$$ (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025).

A key property is that this inductive limit "commutes" with the direct integral: $$\varinjlim_{\alpha \in \Lambda} \int_X^\oplus H_{\alpha,p} d\mu(p) = \int_X^\oplus \left( \varinjlim_{\alpha \in \Lambda} H_{\alpha,p} \right)\,d\mu(p)$$ ensuring the resulting space is itself a locally Hilbert space.

2. Operator Algebras: Decomposable and Diagonalizable Operators

Given the direct integral of locally Hilbert spaces, one is interested in two fundamental classes of bounded operators:

• Decomposable locally bounded operators: For $T \in \mathcal{B}(\mathcal{D}_{\text{loc}})$, $T$ is decomposable if there exists a measurable family $\{T_p\}_{p \in X}$ where $T_p$ is a locally bounded operator on $H_p$, such that

$(Tu)(p) = T_p u(p)$

for $u \in \mathcal{D}_{\text{loc}}$, almost everywhere $p$. This class generalizes the classical decomposable operators for direct integral Hilbert spaces and forms a locally von Neumann algebra—conceptually a projective limit of von Neumann algebras associated to each fiber (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025).

• Diagonalizable locally bounded operators: These are special decomposable operators for which $T_p = f(p)\mathrm{Id}$ on $H_p$, for a measurable essentially bounded function $f: X \to \mathbb{C}$; thus,

$(Tu)(p) = f(p)u(p)$

The set of all such diagonalizable operators forms an abelian locally von Neumann algebra, which can be identified with the algebra of locally essentially bounded measurable functions $E^{\text{loc}}_\text{Bloc}(X,\Sigma,\mu)$ via a normal *-homomorphism $f \mapsto T_f$ (Kulkarni et al., 2 Sep 2024, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025).

Commutant Relationship

Under certain technical conditions (e.g., countable $\Lambda$ or counting measures), the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the decomposable operators: $$\mathfrak{A}_{\text{DIAG}} = (\mathfrak{A}_{\text{DEC}})'$$ This generalizes a fundamental fact from classical direct integral theory to the locally Hilbert context.

3. Rigged Hilbert Space Perspective, Projective and Inductive Limits

The direct integral construction is tightly connected to the duality of inductive (direct) and projective (inverse) limits in the context of contractive families of Hilbert spaces and the theory of rigged Hilbert spaces (RHS). For a directed system $\{H_\alpha\}$ with contraction maps $U_{\beta\alpha}$, the projective limit

$$D = \left\{ (\xi_\alpha)_\alpha \mid U_{\beta\alpha}^* \xi_\beta = \xi_\alpha \right\}$$

captures the "test function" spaces (smooth, regular elements). The inductive limit

$D^\times = \bigcup_\alpha O_\alpha(H_\alpha)$

corresponds to generalized functions/distributions. Both appear naturally at the extremes of the triplet $D \subset H_0 \subset D^\times$ and mirror the fiberwise direct integral picture (Bellomonte et al., 2013, Gheondea, 25 Jul 2025).

4. Functional Models, Spectral Theory, and Applications

With strictly inductive systems of measure spaces, one can construct functional models for locally normal operators as projective limits of multiplication operators, each defined by locally $L^\infty$ functions (Gheondea, 25 Jul 2025): $N = \varprojlim_{\alpha \in \Lambda} N_\alpha,$ where $N_\alpha$ is multiplication by a function $\phi_\alpha \in L^\infty(X_\alpha,\mu_\alpha)$. The associated spectral theorem holds by representing the operator as a direct integral: $N = \int_X^\oplus S_{\varphi(x)} d\mu(x),$ where $S_{\varphi(x)}$ is multiplication by $\varphi(x)$ on the fiber $H(x)$.

Applications include:

• Representation theory and spectral analysis on fractal sets via inductive systems of measure spaces (e.g., the Hata tree-like set) (Gheondea, 25 Jul 2025).
• Quantum mechanics and quantum statistical mechanics—fiberwise decomposition of Hilbert spaces and operators is fundamental in direct integral representations of von Neumann algebras and measurement prescriptions (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025).
• Operator models in locally Hilbert modules and locally $C^*$-algebras, tensor product constructions, and dilation theory (Gheondea, 2015, Gaşpar et al., 2015).
• Reproducing kernel locally Hilbert spaces, and analysis of translation-invariant operators via direct integral decompositions (Herrera-Yañez et al., 2021, Bais et al., 26 Apr 2025).

5. Locally Measure Spaces and Generalizations

The direct integral theory for locally Hilbert spaces is built upon the abstraction of the measure space itself. A "locally measure space" is constructed as a net $(X_\alpha, \Sigma_\alpha, \mu_\alpha)$ with $X = \bigcup_\alpha X_\alpha$, $$\Sigma = \{ E \subset X : E \cap X_\alpha \in \Sigma_\alpha \}$$, and $\mu(E) = \lim_\alpha \mu_\alpha(E \cap X_\alpha)$ (Kulkarni et al., 5 Aug 2025).

This flexible construction facilitates direct integral decompositions where the underlying "base" has local structure, such as spaces built from inductive unions or hierarchically organized measure spaces.

6. Smooth Fields, Bundles, and Differential-Geometric Structures

Direct integrals of locally Hilbert spaces naturally extend to contexts where the fibers vary smoothly. Given a smooth surjective map $p : H \to N$ (typically $N$ a manifold), together with a dense space of sections and compatible connection, one obtains a "smooth field of Hilbert spaces" (Belmonte et al., 2023). Under further conditions (smoothness of transition maps, existence of local trivializations), these fields become Hilbert bundles. Riemannian direct image constructions demonstrate these principles, offering analytic formulas for differentiation along fibers and linking direct integral theory to geometric quantization: $$H(\lambda) = L^2(M_\lambda, E), \quad X F(\lambda) = \int_{M_\lambda} \left( X(f) + [\operatorname{div}X - (\operatorname{div}X \circ p)] f \right) d\mu_x$$

7. Applications in Operator Theory, Quantum Analysis, and Beyond

• Decomposition theory for operator semigroups: Direct integrals enable characterization and asymptotic analysis for globally defined $C_0$-semigroups in terms of component semigroups acting fiberwise; generator and resolvent formulas are direct integrals of the respective fiberwise quantities (Ng, 2019).
• Monotone operator theory: Direct integrals of monotone operators yield frameworks for distributed inclusion and variational problems; direct integral subdifferentials, Moreau envelopes, and proximity operators inherit the relevant properties from the fiber operators/functions (Bùi et al., 2023).
• Abstract sectors in gravitational quantum theory: Sectors labeled by $\alpha$ can be modeled as local Hilbert spaces; the total Hilbert space is organized as a direct integral over sectors, and positivity conditions bound global traces in terms of sector dimensions (Chen, 21 May 2025).

Conclusion

The theory of direct integral of locally Hilbert spaces rigorously generalizes classical direct integral constructions and incorporates the flexibility of inductive limit architectures. It provides a coherent abstract framework for operator fiberization, spectral theory, and algebraic decompositions—encompassing applications from quantum physics to analysis on fractal structures. Key operator classes (decomposable and diagonalizable) are characterized via locally von Neumann algebra structures and commutant dualities, guaranteeing structural properties analogous to the classical setting. The technical machinery—including locally measure spaces, projective/inductive limits, and smooth bundle constructions—connects seamlessly to modern functional analysis, representation theory, and noncommutative geometry.