Decomposable Locally Bounded Operators

• Decomposable locally bounded operators are fiberwise-defined operators on direct integrals of locally Hilbert spaces via measurable fields of locally bounded operators.
• They form a locally von Neumann algebra with operations defined fiberwise and a commutant duality with diagonalizable operators.
• These operators underpin advances in noncommutative geometry, spectral theory, and representation theory in local and nonseparable contexts.

A decomposable locally bounded operator is, broadly, an operator defined on a generalized direct integral of locally Hilbert spaces (or on analogous “fibered” objects such as Lᵖ-direct integrals or vector lattices) that acts via a measurable field of operators, assigning to each fiber a locally bounded operator. This class robustly generalizes classical decomposable operators and is fundamental to the structural theory of operator algebras in non-separable or “local” contexts. Below are the core aspects that define, analyze, and classify decomposable locally bounded operators, synthesized from the research literature with particular reference to the most recent operator-algebraic, functional-analytic, and convex-analytic developments.

1. Definitions and Fiberwise Structure

Let $(X,\Sigma, \mu)$ be a (locally) measure space, and let each $p \in X$ be assigned a locally Hilbert space $\mathcal{D}_p$, typically realized as an inductive limit of an increasing family $\{\mathcal{H}_{p,\alpha}\}_{\alpha \in \Lambda}$ of Hilbert spaces. The direct integral of locally Hilbert spaces is then defined as

$$\mathcal{D}_\mathfrak{d} = \int_X^\oplus \mathcal{D}_p\, d\mu(p)$$

where, for each $u\in \mathcal{D}_\mathfrak{d}$, the function $p\mapsto u(p)$ is required to be measurable with respect to the corresponding fiber structure, and to have certain integrability properties determined by the underlying inductive and direct integral construction (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).

A decomposable locally bounded operator $T$ is then a locally bounded operator on $\mathcal{D}_\mathfrak{d}$ such that there exists a measurable family $\{T_p : \mathcal{D}_p \to \mathcal{D}_p\}_{p\in X}$ of locally bounded operators so that

$(Tu)(p) = T_p u(p) \quad \text{for %%%%9%%%%-a.e. %%%%10%%%%}$

for every $u \in \mathcal{D}_\mathfrak{d}$. The restriction of $T$ to each “stage” $\mathcal{H}_{p,\alpha}$ defines a bounded decomposable operator in the classical sense on the Hilbert space direct integral $\int_X^\oplus \mathcal{H}_{p,\alpha} d\mu$ [(Kulkarni et al., 5 Aug 2025), Eqns (5.3)-(5.4)].

A diagonalizable locally bounded operator is a special case for which $T_p = f(p) \cdot \operatorname{Id}_{\mathcal{D}_p}$ for a measurable function $f: X\to \mathbb{C}$, i.e., $(Tu)(p) = f(p) u(p)$. The set of all such operators forms an abelian von Neumann algebra (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).

2. Algebraic and Lattice-theoretic Structure

The set of all decomposable locally bounded operators on a direct integral $\mathcal{D}_\mathfrak{d}$ forms a locally von Neumann algebra: $$C^*_{\mathfrak{E}, \mathrm{DEC}}(\mathcal{D}_\mathfrak{d}) = \varprojlim_{\alpha\in \Lambda} C^*_{\mathfrak{E}_\alpha,\mathrm{DEC}}\left(\int^\oplus_X \mathcal{H}_{p,\alpha} d\mu\right)$$ where each $C^*_{\mathfrak{E}_\alpha,\mathrm{DEC}}$ is a von Neumann algebra of decomposable operators on a classical Hilbert direct integral, and the inverse/projective limit is taken in the locally convex topology (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).

The operations in this algebra are performed fiberwise:

• $(T+S)_p = T_p + S_p$
• $(\lambda T)_p = \lambda T_p$
• $(TS)_p = T_p S_p$
• $(T^*)_p = (T_p)^*$

Analogously, the set of all diagonalizable locally bounded operators forms an abelian locally von Neumann algebra. It is structurally identified with a suitable $L^\infty(X, \mu)$-type algebra, acting by fiberwise scalar multiplication.

A defining feature ((Kulkarni et al., 5 Aug 2025), Thm 5.7) is that the algebra of decomposable locally bounded operators is exactly the commutant of the diagonalizable algebra: $$C^*_{\mathfrak{E},\mathrm{DEC}}(\mathcal{D}_\mathfrak{d}) = \left(C^*_{\mathfrak{E},\mathrm{DIAG}}(\mathcal{D}_\mathfrak{d})\right)'$$ which generalizes the classical von Neumann double commutant theorem for direct integral von Neumann algebras.

3. Concrete Examples and Structural Dichotomy

• Diagonalizable Example: $T$ defined by $(Tu)(p) = f(p) u(p)$ for $f\in L^\infty(X,\mu)$ is diagonalizable and decomposable.
• Decomposable Non-diagonalizable Example: A direct integral with a single non-zero fiber given by $T_1(e_k) = k e_k$ on $\ell^2(\mathbb{N})$, vanishing elsewhere, defines a decomposable operator that is not diagonalizable, since no scalar function $f$ can replicate the variable $k$ on $\ell^2(\mathbb{N})$ [(Kulkarni et al., 5 Aug 2025), Ex. 3.2].
• Operators Not Decomposable: If $T$ does not admit any measurable field of operators acting fiberwise due to irregularity in dependence on $p$, it is not decomposable [(Kulkarni et al., 5 Aug 2025), Ex. 3.4].

This structural dichotomy demarcates the operator algebra into a noncommutative (decomposable) core and an abelian (diagonalizable) component, which critically shapes representation theory and spectral analysis in the locally Hilbert space context.

4. The Role in Noncommutative and Fibered Analysis

Decomposable locally bounded operators generalize classical constructions in several analytic settings:

• Von Neumann Algebras: The mutual commutant identity between decomposable and diagonalizable operators mirrors direct integral results and underpins the extension to nonseparable or “local” operator algebras (Kulkarni et al., 5 Aug 2025, Kulkarni et al., 5 Aug 2025, Kulkarni et al., 2 Sep 2024).
• L^p-Direct Integrals & Operator-valued Multipliers: The decomposable framework underpins the definition of operator-valued and Fourier multipliers on group von Neumann algebras (Arhancet et al., 2017, Arhancet et al., 2022), whereby operators act fiberwise or with respect to group representations, with decomposition ensuring positivity, complete boundedness, and compatibility with harmonic analysis structures.
• General Banach Structure: Even beyond Hilbert spaces, analogous decompositions arise in $L^p$-direct integrals of Banach spaces (Evseev et al., 2019), where boundedness of a decomposable operator is characterized by essential supremum conditions on associated measurable families.

5. Characterizations and Dualities

Commutant Duality

The explicit commutant structure,

$$C^*_{\mathfrak{E},\mathrm{DEC}}(\mathcal{D}_\mathfrak{d}) = ( C^*_{\mathfrak{E},\mathrm{DIAG}}(\mathcal{D}_\mathfrak{d}) )'$$

can be heuristically understood as follows:

• Any operator commuting with all diagonalizable (multiplication) operators must leave each fiber invariant, i.e., act fiberwise—thus is decomposable.
• Conversely, each decomposable operator trivially commutes with every diagonalizable one due to fiberwise action.

Projective and Inductive Limit Structure

The construction of direct integrals and the corresponding operator algebras is realized via projective and inductive limits over the directed system $\Lambda$: $$\mathcal{D}_\mathfrak{d} = \varinjlim_{\alpha\in\Lambda} \int_X^\oplus \mathcal{H}_{p,\alpha} d\mu, \qquad C^*_{\mathfrak{E},\mathrm{DEC}}(\mathcal{D}_\mathfrak{d}) = \varprojlim_{\alpha\in\Lambda} C^*_{\mathfrak{E}_\alpha,\mathrm{DEC}}\left(\int_X^\oplus \mathcal{H}_{p,\alpha} d\mu\right)$$ This captures the “local” nature of the operator algebra and provides a path from finite-stage von Neumann algebraic theory to the global, locally Hilbert space setting.

6. Relevance to Modern Operator Theory

The recognition of decomposable locally bounded operators as central algebraic objects connects several areas:

• Noncommutative geometry and quantum probability: Fibered or “local” operator algebras arise naturally in continuous fields of C*-algebras, quantum field theory, and the theory of noncommutative Fock spaces.
• Spectral theory: The spectral calculus of locally bounded decomposable operators directly generalizes classical direct integral techniques, with the abelian subalgebra playing the role of the “diagonal” spectrum.
• Representation theory: Such operators provide the basic building blocks for decomposing representations of locally C*- or von Neumann algebras in terms of measurable fields.

7. Summary Table

Operator Type	Definition (Fiberwise)	Algebraic Structure
Decomposable	$(Tu)(p) = T_p u(p)$	Locally von Neumann algebra
Diagonalizable	$(Tu)(p) = f(p) u(p)$	Abelian locally von Neumann algebra
Commutant Relation	$T$ commutes with all diagonalizable $\iff$ $T$ is decomposable	$(\mathrm{DIAG})' = \mathrm{DEC}$

• (Kulkarni et al., 5 Aug 2025) Direct integral of locally Hilbert spaces
• (Kulkarni et al., 5 Aug 2025) Direct integral of locally Hilbert spaces over a locally measure space
• (Kulkarni et al., 2 Sep 2024) Direct Integral and Decompositions of Locally Hilbert spaces

Conclusion

Decomposable locally bounded operators are characterized as operators acting fiberwise on the direct integral of locally Hilbert spaces via measurable families of locally bounded operators. They constitute a locally von Neumann algebra whose commutant is the abelian algebra of all diagonalizable (multiplication) operators; under reasonable measure-theoretic or index set conditions, this mutual commutant structure mirrors classical results for direct integrals of Hilbert spaces. Their systematic analysis using projective/inductive limits provides foundational machinery for the paper of representations, spectral theory, and harmonic analysis on generalized operator algebras in the “local” setting.