Abstract
We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.
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Acknowledgements
The author is grateful to Professors Sergio Albeverio and Andreas Eberle, and to Dr. Kohei Suzuki, for fruitful conversations on the subject of the present work, and for respectively pointing out the references [1, 13], and [3, 20]. Finally, he is especially grateful to an anonymous Reviewer for their very careful reading and their suggestions which improved the readability of the paper.
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A large part of this work was written while the author was a member of the Institut für angewandte Mathematik of the University of Bonn. Research supported by the Collaborative Research Center 1060. The author gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) grant F65, and by the European Research Council (ERC, grant No. 716117, awarded to Prof. Dr. Jan Maas).
Appendix
Appendix
The theory of direct integrals of Banach spaces is inherently more sophisticated than the corresponding theory for Hilbert spaces. We discuss here an irreducible minimum after [17, Ch.s 5-7] and especially [9, §3]. For simplicity, we restrict ourselves to the case of σ-finite (not necessarily complete) indexing spaces \((Z,\mathcal {Z},\nu )\).
A decomposition \(\left (Z_{\alpha }\right )_{\alpha \in \mathrm {A}}\) of \((Z,\mathcal {Z},\nu )\) is a family of subsets Zα ⊂ Z so that
Definition 1.1 (Measurable fields, cf. [17, §6.1, p. 61f.] and [9, §3.1])
Let \((Z,\mathcal {Z},\nu )\) be a σ-finite measure space, and V be a real linear space. A ν-measurable family of semi-norms on V is a family (q∥⋅∥ζ)ζ∈Z so that
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\(\left \lVert { \cdot }\right \rVert _{\zeta }\) is a semi-norm on V for every ζ ∈ Z;
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the map \(\zeta \mapsto \left \lVert v\right \rVert _{\zeta }\) is ν-measurable for every v ∈ V.
Letting Yζ denote the Banach completion of \(V/\ker \left \lVert { \cdot }\right \rVert _{\zeta }\), we say that a vector field \(u\in {\prod }_{\zeta \in Z} Y_{\zeta }\) is ν-measurable if, for each \(B\in \mathcal {Z}\) with \(\nu B<\infty \), there exists a sequence (un)n of simple V-valued vector fields on B so that \(\lim \|{u_{\zeta }-u_{n,\zeta }}\|_{\zeta }=0\) ν-a.e. on B.
A family \(\left (Y_{\zeta }\right )_{\zeta \in Z}\) of Banach spaces Yζ is a ν-measurable field of Banach spaces if there exist
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a decomposition \(\left (Z_{\alpha }\right )_{\alpha \in \mathrm {A}}\) of \((Z,\mathcal {Z},\nu )\) consisting of sets of finite ν-measure;
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a family of real linear spaces \(\left (Y^{\alpha }\right )_{\alpha \in \mathrm {A}}\);
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for each α ∈A, a ν-measurable family of norms \(\left \lVert { \cdot }\right \rVert _{\zeta }\) on Yα,
so that, for each α ∈A and each ζ ∈ Zα, the space Yζ is the completion of \((Y^{\alpha },\left \lVert { \cdot }\right \rVert _{\zeta })\).
Extending the above definition of ν-measurability, we say that \(u\in {\prod }_{\zeta \in Z} Y_{\zeta }\) is ν-measurable if (and only if) the restriction of u to each Zα is ν-measurable.
Let \(p\in [1,\infty ]\). A ν-measurable vector field u is called Lp(ν)-integrable if \(\left \lVert u\right \rVert _{p}:= \big \lVert (\zeta \mapsto \left \lVert u_{\zeta }\right \rVert _{\zeta }) \big \rVert _{L^{p}(\nu )}\) is finite. Two Lp(ν)-integrable vector fields u, v are ν-equivalent if ∥u − v∥p = 0.
The space Yp of equivalence classes of Lp(ν)-integrable vector fields modulo ν-equivalence, endowed with the non-relabeled quotient norm \(\left \lVert { \cdot }\right \rVert _{p}\), is a Banach space [9, Prop. 3.2], called the Lp-direct integral of ζ↦Yζ and denoted by
The following is a generalization of Proposition 2.25 to direct integrals of Lp-spaces. Recall (16).
Proposition 1.2
Let \((X,\mathcal {X},\mu )\) be σ-finite standard, \((Z,\mathcal {Z},\nu )\) be σ-finite countably generated, and \(\left (\mu _{\zeta }\right )_{\zeta \in Z}\) be a separated pseudo-disintegration of μ over ν. Further let \(\mathcal {A}\) be the lattice algebra of all real-valued μ-integrable simple functions on \((X,\mathcal {X})\). Then, for every \(p\in [1,\infty )\), the map
extends to an isomorphism of Banach lattices \(\iota _{p}\colon L^{p}(\mu ) \rightarrow Y_{p}\).
A proof of the above Proposition 1.2 is quite similar to that of Proposition 2.25, and therefore it is omitted. Alternatively, a proof may be adapted from [9, §4.2], having care that:
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the algebra \(\mathcal {A}\) corresponds to the vector lattice V in [9, p. 694];
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the order on Yp is defined analogously to Remark 2.20, cf. [9, p. 694];
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the map ι corresponds to the map defined in [9, Eqn. (4.6)];
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the surjectivity of ιp follows as in [9, p. 696] since it only depends on the disintegration being separated. In the terminology and notation of [9], this is accounted by the fact that the decomposition β satisfies [9, Thm. 4.2(2)].
As an obvious corollary to Proposition 1.2, we obtain that the direct integral of Hilbert spaces H in Eq. 18 with underlying space of measurable vector fields generated by \(\mathcal {A}\) is identical to Y2 as in Eq. 50. The specification of the underlying space of ν-measurable vector fields is necessary in light of Remark 2.23.
Proof of Lemma 3.8
Retain the notation established in Section 3.1 and in the proof of Theorem 3.4. Firstly, note that L1(μ) is, trivially, an \(L^{\infty }(\mu _{0})\)-module, and \(\mathcal {D}(E)\) is an \(L^{\infty }(\mu _{0})\)-module too, by Definition 3.1(d). As in Section 3.1, let p be the quotient map of Eq. 27. For \(u\in L^{\infty }(\nu )\) denote by \(p^{*}u\in L^{\infty }(\mu _{0})\) the pullback of u via p. Setting u.: f↦p∗u ⋅ f defines an action of \(L^{\infty }(\nu )\) on L2(μ) and \(\mathcal {D}(E)\). Thus, since the spaces \((X,\mathcal {X}_{0},\mu _{0})\) and \((Z,\mathcal {Z},\nu )\) have the same measure algebra by construction of Z, here and in the following we may replace \(L^{\infty }(\mu _{0})\)-modularity with \(L^{\infty }(\nu )\)-modularity.
Let now \(A\in \mathcal {X}_{0}\). Since A is E-invariant, then and
by Definition 3.1. Replacing f with in Eq. 13, and applying (51) and again (13) yields
which is readily extended to \(f,g\in \mathcal {D}(E)\) by approximation. Then, Eq. 52 shows that \(f\mapsto {\Gamma }(f,g)\colon \mathcal {D}(E)_{1}\rightarrow L^{1}(\mu )\) is, for every fixed \(g\in \mathcal {D}(E)\), a bounded \(L^{\infty }(\nu )\)-modular operator in the sense of [17, §5.2]. By Step 1 in the proof of Theorem 3.4(iii), \(\mathcal {D}(E)_{1}\) is a countably generated direct integral of Banach spaces, thus we may apply [17, Thm. 9.1] to obtain, for every fixed \(g\in \mathcal {D}(E)\), a direct integral decomposition
for some family of bounded operators \({\Gamma }_{\zeta ,g}\colon \mathcal {D}(E_{\zeta })_{1}\rightarrow L^{1}(\mu _{\zeta })\). Let \(\mathcal {C}\) be a core for \((E,\mathcal {D}(E))\) underlying the construction of the direct integral representation of \((E,\mathcal {D}({E}))\) as in Step 1 in the proof of Theorem 3.4. It follows by symmetry of Γ that Γζ, g(f) = Γζ, f(g) for every \(f,g\in \mathcal {C}\) and ν-a.e. ζ ∈ Z. In particular, the assignment g↦Γζ, g is linear on \(\mathcal {C}\subset \mathcal {D}({E_{\zeta }})\) for ν-a.e. ζ ∈ Z. A symmetric bilinear map is then induced on \(\mathcal {C}^{\oplus {2}}\) by setting Γζ: (f, g)↦Γζ, g(f).
Thus, finally, it suffices to show Eq. 13 for Γζ and \((E_{\zeta },\mathcal {D}(E_{\zeta }))\) for ν-a.e. ζ ∈ Z with \(f,g,h\in \mathcal {C}\), which is readily shown arguing by contradiction, analogously to the proof of the claim in Step 4 of Theorem 3.4. □
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Dello Schiavo, L. Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications. Potential Anal 58, 573–615 (2023). https://doi.org/10.1007/s11118-021-09951-y
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DOI: https://doi.org/10.1007/s11118-021-09951-y