# Modular Nuclearity: A Generally Covariant Perspective

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## Abstract

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## 1. Introduction

## 2. The Modular ${\ell}^{p}$-Condition

#### 2.1. Modular Operators and ${\ell}^{p}$-Conditions

**Lemma 2.1.**If ${\Xi}_{2}:{\mathcal{B}}_{1}\to {\mathcal{B}}_{2}$ and ${\Xi}_{3}:{\mathcal{B}}_{1}\to {\mathcal{B}}_{3}$ are bounded linear maps such that $\parallel {\Xi}_{3}(b)\parallel \le \parallel {\Xi}_{2}(b)\parallel $ for all $b\in {\mathcal{B}}_{1}$, then $\parallel {\Xi}_{3}{\parallel}_{p}\le {\parallel {\Xi}_{2}\parallel}_{p}$ for all $p>0$.

**Proof.**The estimate allows us to define a linear map B from the range of ${\Xi}_{2}$ to the range of ${\Xi}_{3}$ such that ${\Xi}_{3}(b)=B{\Xi}_{2}(b)$ and $\parallel B\parallel \le 1$. B has an extension to ${\mathcal{B}}_{2}$ with $\parallel B\parallel \le 1$, by the Hahn-Banach Theorem, and hence $\parallel {\Xi}_{3}{\parallel}_{p}=\parallel B{\Xi}_{2}{\parallel}_{p}\le {\parallel {\Xi}_{2}\parallel}_{p}$. ☐

**Remark 2.2.**We will also need to consider real linear ${\ell}^{p}$ maps between real Banach spaces, which are defined in a completely analogous way. In this context we denote the n’th approximation number by ${\alpha}_{n}^{\mathbb{R}}(\Xi )$ and the corresponding quasi-norms ${\parallel \Xi \parallel}_{\mathbb{R},p}^{p}={\sum}_{n=0}^{\infty}{\alpha}_{n}^{\mathbb{R}}{(\Xi )}^{p}$. The estimates (2.5) and (2.6) hold also in this case.

**Lemma 2.3.**Let ${\mathcal{B}}_{1},{\mathcal{B}}_{2}$ be Banach spaces, and $0<p\le 1$. Then

**Proof.**The first inclusion and the bound (2.10) is proven in [34, Prop. 8.4.2] for $0<p\le 1$, see also [35] for the extension to $p>1$.

#### 2.2. Definition of the Modular ${\ell}^{p}$-Condition

**Definition 2.4**(Modular ${\ell}^{p}$-condition). Let M be a globally hyperbolic spacetime and ω a state on $\mathcal{A}(M)$. We say that ω satisfies the modular ${\ell}^{p}$-condition if and only if for all $\alpha \in (0,\frac{1}{2})$, $p>0$ and all compact inclusions $\iota :O\to M$ and $\tilde{\iota}:\tilde{O}\to O$ the maps ${\Xi}_{\tilde{O},O;{\iota}^{*}\omega}^{(\alpha )}$ are ${\ell}^{p}$.

#### 2.3. Stability Properties of the Modular ${\ell}^{p}$-Condition

**Lemma 2.5.**If $\psi :\tilde{M}\to M$ is a morphism and ω a state on $\mathcal{A}(M)$ which satisfies the modular ${\ell}^{p}$-condition, then the pull-back $\tilde{\omega}:=\mathcal{A}{(\psi )}^{*}\omega $ also satisfies the modular ${\ell}^{p}$-condition, because for any compact inclusions $\iota :O\to \tilde{M}$ and $\tilde{\iota}:\tilde{O}\to O$ we have ${\Xi}_{\tilde{O},O;{\iota}^{*}\tilde{\omega}}^{(\alpha )}={\Xi}_{\tilde{O},O;{(\psi \xb0\iota )}^{*}\omega}^{(\alpha )}$.

**Theorem 2.6**(Löwner’s Theorem). Let $I=(a,b)\subset \mathbb{R}$ be an open interval, where $b=\infty $ is allowed, and let $f:I\to \mathbb{R}$ be a continuous function. Then the following two statements are equivalent:

- (a)
- There is a holomorphic function F on the upper half complex plane such that $\mathrm{Im}(F(z))>0$ and which has f as a continuous boundary value on I.
- (b)
- For all self-adjoint (possibly unbounded) operators A, B on a Hilbert space $\mathcal{H}$ with $a<A\le B<b$ (or $a<A\le B$ when $b=\infty $) on the form domain of B, we also have $f(A)\le f(B)$ on the intersection of the form domains of $f(A)$ and $f(B)$.

**Proof.**In the standard version of Löwner’s Theorem one replaces the second statement by a weaker one, where only bounded operators A and B with spectrum in I are allowed [39]. We will not repeat the proof of that result here, but only show that the weaker version of the second statement implies the stronger one. By a translation we may assume that $a=0$, so $0<A\le B$. For $n\in \mathbb{N}$ we set ${a}_{n}:={n}^{-1}$ and ${b}_{n}:=b-{n}^{-1}$, or ${b}_{n}:=n$ when $b=\infty $. We let ${E}_{n}$ and ${F}_{n}$ be the spectral projections for A and B, respectively, onto $[{a}_{n},{b}_{n}]$ and we fix $c\in (0,b)$. We then define

**Remark 2.7.**When f is operator monotonic, it is monotonically increasing. If it has a continuous extension to the lower boundary a of the interval I with $f(a)\le 0$, then the second statement can be extended to operators $A,B$ such that $a\le A\le B<b$ (or $a\le A\le B$), in which case $f(A)\le f(B)$ on the form domain of B. Indeed, the eigenspace of $\sqrt{B+a}$ of eigenvalue 0 is contained in the eigenspace of $\sqrt{A+a}$ of eigenvalue 0, so both operators act the same way on this subspace and it remains to consider the orthogonal complement. There, however, we may repeat the proof of Theorem 2.6, supplementing the last line with the remark that $\langle \psi ,f(A)\psi \rangle \le {lim}_{m\to \infty}\langle \psi ,{E}_{m}f(A){E}_{m}\psi \rangle $ (because $f(a)\le 0$), and ${lim}_{m\to \infty}\psi $ is in the form domain of $f(A)$, because $f(A)$ is semi-bounded from below.

**Corollary 2.8.**Let $I=(a,b)$ as in Theorem 2.6 and let $f:I\to \mathbb{R}$ be operator monotonic with a continuous extension to a such that $f(a)\le 0$. Let A be a self-adjoint operator on a Hilbert space $\mathcal{H}$ and B on a subspace ${\mathcal{H}}^{\prime}$. When $a\le A<b$ and $a\le B<b$ (or $a\le A$ and $a\le B$ when $b=\infty $), and when $A\le B$ on the form domain of B, then $f(A)\le f(B)$ on the form domain of $f(B)$.

**Proof.**We let P denote the orthogonal projection in $\mathcal{H}$ onto ${\mathcal{H}}^{\prime}$ and for any $n\in \mathbb{N}$ we let ${E}_{n}$ be the spectral projection of A onto $[a,b-{n}^{-1}]$ (or $[a,a+n]$ when $b=\infty $). We set ${A}_{n}:={E}_{n}A+a(1-{E}_{n})$, which is a bounded operator with spectrum in $[a,b)$. For any $\u03f5>0$ we then note that

**Lemma 2.9.**Let $\mathcal{B}\subset \mathcal{A}$ be an inclusion of ${C}^{*}$-algebras and let ω be a state on $\mathcal{A}$ with restriction λ to $\mathcal{B}$. For all $b\in \mathcal{B}$ and $\alpha \in [0,\frac{1}{2}]$ we then have

**Proof.**Let P denote the orthogonal projection in ${\mathcal{H}}_{\omega}$ onto $\overline{{\pi}_{\omega}(\mathcal{B}){\Omega}_{\omega}}$, so we may identify the GNS-representation of λ as ${\pi}_{\lambda}:=P{\pi}_{\omega}{|}_{\mathcal{B}}P$, ${\mathcal{H}}_{\lambda}:=P{\mathcal{H}}_{\omega}$ and ${\Omega}_{\lambda}:={\Omega}_{\omega}$. Let ${Q}_{\omega}$ and ${Q}_{\lambda}$ be the orthogonal projections onto ${\mathcal{H}}_{\omega}^{\prime}$ and ${\mathcal{H}}_{\lambda}^{\prime}$, where we extend ${Q}_{\lambda}$ to ${\mathcal{H}}_{\omega}$ by setting ${Q}_{\lambda}={Q}_{\lambda}P$. Note that $P\in {\pi}_{\omega}{(\mathcal{B})}^{\prime}$ and ${\pi}_{\lambda}{(\mathcal{B})}^{\prime}={(P{\pi}_{\omega}(\mathcal{B})P)}^{\prime}\supset P{\pi}_{\omega}{(\mathcal{A})}^{\prime}P$. It follows that ${\mathcal{H}}_{\lambda}^{\prime}\supset P{\mathcal{H}}_{\omega}^{\prime}$ and hence $P{Q}_{\omega}={Q}_{\lambda}P{Q}_{\omega}={Q}_{\lambda}{Q}_{\omega}$ and ${Q}_{\omega}P={Q}_{\omega}{Q}_{\lambda}$. Hence,

**Proposition 2.10.**Let ${\omega}_{1}$ and ${\omega}_{2}$ be two states on $\mathcal{A}(M)$ and $\omega ={r}_{1}{\omega}_{1}+{r}_{2}{\omega}_{2}$ for some ${r}_{1},{r}_{2}>0$ with ${r}_{1}+{r}_{2}=1$. Then ω satisfies the modular ${\ell}^{p}$-condition if both ${\omega}_{i}$ do. Moreover, for any two compact inclusions $\iota :O\to M$ and $\tilde{\iota}:\tilde{O}\to O$ we have

**Proof.**Denote the pull-backs of the states to O by ${\lambda}_{i}:={\iota}^{*}{\omega}_{i}$ and note that $\lambda :={\iota}^{*}\omega ={r}_{1}{\lambda}_{1}+\phantom{\rule{3.33333pt}{0ex}}{r}_{2}{\lambda}_{2}$. Let $\mathcal{H}:={\mathcal{H}}_{{\lambda}_{1}}\oplus {\mathcal{H}}_{{\lambda}_{2}}$ and $\Omega :=\sqrt{{r}_{1}}{\Omega}_{{\lambda}_{1}}\oplus \sqrt{{r}_{2}}{\Omega}_{{\lambda}_{2}}$. By construction, the modular operator for $\mathcal{M}:={\pi}_{{\lambda}_{1}}{(\mathcal{A}(O))}^{\u2033}\oplus {\pi}_{{\lambda}_{2}}{(\mathcal{A}(O))}^{\u2033}$ and Ω is $\Delta :={\Delta}_{{\lambda}_{1}}\oplus {\Delta}_{{\lambda}_{2}}$. For the map

**Corollary 2.11.**Let ${\mathcal{B}}_{1}\subset {\mathcal{B}}_{2}\subset {\mathcal{A}}_{2}\subset {\mathcal{A}}_{1}$ be inclusions of ${C}^{*}$-algebras and let ${\omega}_{1}$ be a state on ${\mathcal{A}}_{1}$ with restriction ${\omega}_{2}$ to ${\mathcal{A}}_{2}$. For all $\alpha \in [0,\frac{1}{2}]$, $p>0$, the maps

**Proof.**From Lemma 2.9 we have the estimate $\parallel {\Xi}_{1}^{(\alpha )}(b)\parallel \le \parallel {\Xi}_{2}^{(\alpha )}(b)\parallel $ for all $b\in {\mathcal{B}}_{1}$. Since the injection ${\mathcal{B}}_{1}\subset {\mathcal{B}}_{2}$ is also bounded with norm 1, the estimate follows immediately from Lemma 2.1. ☐

**Lemma 2.12.**Given any compact inclusions $\iota :O\to M$ and $\tilde{\iota}:\tilde{O}\to O$ there is a regular Cauchy pair $(\tilde{V},V)$ in M such that $\overline{\tilde{O}}\subset D(\tilde{V})$ and $\overline{V}\subset O$.

**Proof.**Let $p\in O$ be a point in the causal complement of $\overline{\tilde{O}}$ and let $\mathcal{C}$ be a smooth space-like Cauchy surface for O containing p. Let $J(\overline{\tilde{O}})$ denote the union of the causal future and past of $\overline{\tilde{O}}$ and note that $K:=J(\overline{\tilde{O}})\cap \mathcal{C}$ is compact with a non-empty complement in $\mathcal{C}$. We may then choose relatively compact open subsets $\tilde{V},V$ in $\mathcal{C}$ such that $K\subset \tilde{V}$, $\overline{\tilde{V}}\subset V$ and $\overline{V}$ has a non-empty complement in $\mathcal{C}$, i.e., $(\tilde{V},V)$ is a regular Cauchy pair in O. It follows from Lemma 2.4 in [42] that $(\tilde{V},V)$ is also a regular Cauchy pair in M and the desired inclusions follow from the construction. ☐

**Theorem 2.13.**Assume that the theory $\mathbf{A}$ satisfies the time-slice axiom and let ${M}_{1}$ and ${M}_{2}$ be globally hyperbolic spacetimes with diffeomorphic Cauchy surfaces. Given compact inclusions $\iota :O\to {M}_{1}$ and $\tilde{\iota}:\tilde{O}\to O$ and a Cauchy surface ${\mathcal{C}}_{2}$ of ${M}_{2}$ there is a regular Cauchy pair $({\tilde{V}}_{2},{V}_{2})$ in ${M}_{2}$, contained in ${\mathcal{C}}_{2}$, and a chain of Cauchy morphisms

**Proof.**By the time-slice axiom we have $\mathcal{A}(W)=\mathcal{A}(D(W))$ for any causally convex region $W\subset M$ in any globally hyperbolic spacetime M. Using Lemma 2.12 we find a regular Cauchy pair $({\tilde{V}}_{1},{V}_{1})$ in ${M}_{1}$ such that $\overline{\tilde{O}}\subset D({\tilde{V}}_{1})$ and $\overline{{V}_{1}}\subset O$. Theorem 3.4 in [42] proves the existence of the Cauchy pair $({\tilde{V}}_{2},{V}_{2})$ and a chain of Cauchy morphisms ${\psi}_{1},{\chi}_{1},{\chi}_{2},{\psi}_{2}$ such that the isomorphism ν satisfies

**Remark 2.14.**Suppose we are given sets of states ${\mathcal{S}}_{i}$ on $\mathcal{A}({M}_{i})$ such that all states in ${\mathcal{S}}_{2}$ satisfy the modular ${\ell}^{p}$-condition and all spacetime deformations as in Theorem 2.13 map all states in ${\mathcal{S}}_{1}$ into ${\mathcal{S}}_{2}$. Then it is clear from the theorem that all states in ${\mathcal{S}}_{1}$ also satisfy the modular ${\ell}^{p}$-condition. This argument will be applied in Section 5 to the sets of quasi-free Hadamard states of a free scalar field. It is then sufficient to consider only ultra-static spacetimes ${M}_{2}$, because they already cover all possible diffeomorphism classes of Cauchy surfaces.

## 3. Nuclearity Conditions and Second Quantization

#### 3.1. ${\ell}^{p}$-Conditions and Bosonic Second Quantization

**Lemma 3.1.**Let $H\subset \mathcal{H}$ be a closed real subspace. Then

- (a)
- Ω is cyclic for $\mathcal{M}(H)$ if and only if $H+iH\subset \mathcal{H}$ is dense.
- (b)
- Ω is separating for $\mathcal{M}(H)$ if and only if $H\cap iH=\left\{0\right\}$.
- (c)
- The map $H\mapsto \mathcal{M}(H)$ preserves inclusions.
- (d)
- $\mathcal{M}{(H)}^{\prime}=\mathcal{M}(\stackrel{\xb0}{H})$.

**Theorem 3.2.**[28] Let $H,X$ be as above, satisfying the following two additional assumptions:

- (a)
- There exists a conjugation Γ on $\mathcal{H}$ which commutes with ${X}_{1}$, and two closed complex linear subspaces ${\mathcal{K}}_{\pm}\subset \mathcal{H}$, such that $\Gamma {\mathcal{K}}_{\pm}={\mathcal{K}}_{\pm}$ and$$\begin{array}{cc}\hfill H& ={\Gamma}^{+}{\mathcal{K}}_{+}+{\Gamma}^{-}{\mathcal{K}}_{-}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$
- (b)
- Denoting the (complex linear) projections onto ${\mathcal{K}}_{\pm}$ by ${E}_{\pm}$, the operators ${X}_{1}{E}_{\pm}\in \mathcal{B}(\mathcal{H})$ are trace class and satisfy $\parallel {X}_{1}{E}_{\pm}\parallel <1$.

**Theorem 3.3.**In the notations above, the following hold true:

- (a)
- If ${X}_{1}{E}_{H}$ is ${\ell}_{\mathbb{R}}^{p}(\mathcal{H})$ for some $0<p\le 1$, and $\parallel {X}_{1}{E}_{H}\parallel <1$, then ${\Xi}_{H,{X}_{1}}$ is p-nuclear and ${\ell}^{q}$ for $q>p/(1-p)$. In particular, if the assumption holds for all $p>0$, then ${\Xi}_{H,{X}_{1}}$ is ${\ell}^{q}$ for all $q>0$.
- (b)
- $\parallel {X}_{1}{E}_{H}{\parallel}_{\mathbb{R},p}\le \sqrt{e}\phantom{\rule{0.166667em}{0ex}}{2}^{1/p}\phantom{\rule{0.166667em}{0ex}}{\parallel {\Xi}_{H,{X}_{1}}\parallel}_{p}$ for all $p>0$.

**Lemma 3.4.**

- (a)
- Let Γ be a conjugation on $\mathcal{H}$. Then a closed real subspace $H\subset \mathcal{H}$ is of the form $H={\Gamma}^{+}{\mathcal{K}}_{+}+{\Gamma}^{-}{\mathcal{K}}_{-}$ with two closed complex linear subspaces ${\mathcal{K}}_{\pm}\subset \mathcal{H}$ which are invariant under Γ if and only if $\Gamma H=H$.
- (b)
- If H is a closed real subspace as in $(a)$, the real orthogonal projection ${E}_{H}$ onto H is related to the complex orthogonal projections ${E}_{\pm}$ onto ${\mathcal{K}}_{\pm}$ by$$\begin{array}{c}\hfill {E}_{H}={\Gamma}^{+}{E}_{+}+{\Gamma}^{-}{E}_{-}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$

**Proof.**$(a)$ Suppose H has the form $H={\Gamma}^{+}{\mathcal{K}}_{+}+{\Gamma}^{-}{\mathcal{K}}_{-}$ as described. Then, as ${\Gamma}^{2}=1$, it follows immediately that $\Gamma H=H$.

**Lemma 3.5.**Let ${T}_{\pm}$ be two bounded complex linear operators, and Γ a conjugation that commutes with both of them. Define the real linear operator

- (a)
- There holds the norm equality$$\begin{array}{c}\hfill \parallel T\parallel =max\{\parallel {T}_{+}\parallel ,\parallel {T}_{-}\parallel \}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$
- (b)
- Let $p>0$. Then $T\in {\ell}_{\mathbb{R}}^{p}(\mathcal{H})$ if and only if ${T}_{\pm}\in {\ell}^{p}(\mathcal{H})$, and in this case, the corresponding ${\ell}^{p}$-quasi-norms satisfy the bounds$$\begin{array}{cc}\hfill {\parallel T\parallel}_{\mathbb{R},p}& \le {c}_{p}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\parallel {T}_{+}{\parallel}_{p}+{\parallel {T}_{-}\parallel}_{p}\hfill \end{array}$$$$\begin{array}{cc}\hfill \parallel {T}_{\pm}{\parallel}_{p}& \le {c}_{p}^{\prime}\phantom{\rule{0.166667em}{0ex}}{\parallel T\parallel}_{\mathbb{R},p}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$

**Proof.**$(a)$ The proof of (3.9) is based on the fact that for a conjugation Γ on $\mathcal{H}$ and two arbitrary vectors $\psi ,\xi \in \mathcal{H}$, there always holds

**Proof.**$(a)$ As explained above, we first need to account for the possibility that there is no conjugation Γ such that $[\Gamma ,{X}_{1}]=0$ and $\Gamma H=H$. We therefore start with a construction to introduce some additional structure. Let Γ be a conjugation on $\mathcal{H}$, and consider

#### 3.2. Second Quantization of Modular Operators

**Lemma 3.6.**Let $H\subset \mathcal{H}$ be a standard subspace. Then the modular data $J,\Delta $ of $(\mathcal{M}(H),\Omega )$ are related to ${J}_{H},{\Delta}_{H}$ by second quantization:

**Theorem 3.7.**Let $\tilde{H}\subset H\subset \mathcal{H}$ be an inclusion of closed real subspaces, and $0<\alpha <\frac{1}{2}$. If $\parallel {\Delta}_{H}^{\alpha}{E}_{\tilde{H}}\parallel <1$ and ${\Delta}_{H}^{\alpha}{E}_{\tilde{H}}$ is ${\ell}_{\mathbb{R}}^{p}(\mathcal{H})$ for all $p>0$, then ${\Xi}_{\tilde{H},{\Delta}_{H}^{\alpha}}$ (3.3) is ${\ell}^{p}$ for all $p>0$.

**Proof.**In view of the split (3.22), the Bose Fock space over $\mathcal{H}$ has the form

**Lemma 3.8.**Let $\tilde{H}\subset H$ be an inclusion of closed real subspaces and $0<\alpha <\frac{1}{2}$. Then $\parallel {\Delta}_{H}^{\alpha}{E}_{\tilde{H}}\parallel \le 1$, and $|{\Delta}_{H}^{\alpha}{E}_{\tilde{H}}|<1$ whenever $\tilde{H}\cap \stackrel{\xb0}{H}=\left\{0\right\}$. If in addition ${\Delta}_{H}^{\alpha}{E}_{\tilde{H}}$ is compact (or even ${\ell}^{p}$ for some $p>0$), then

**Proof.**Let $\chi ,\eta \in \mathcal{H}$ and $h:={E}_{H}\eta $. As $H=\mathrm{dom}{\Delta}_{H}^{1/2}$, it follows that the function $f(z):=\langle \chi ,{\Delta}_{H}^{-iz}h\rangle $ is analytic on the strip $0<\mathrm{Im}(z)<\frac{1}{2}$ and continuous on the closure of this strip. In view of $|f(z)|\le \parallel \chi \parallel \parallel {\Delta}_{H}^{\mathrm{Im}(z)}h\parallel $ we see furthermore that f is bounded. Moreover, on the boundary we have $|f(t)|\le \parallel \chi \parallel \parallel h\parallel $, and $|f(t+\frac{i}{2})|\le \parallel \chi \parallel \parallel {\Delta}_{H}^{1/2}h\parallel =\parallel \chi \parallel \parallel {J}_{H}h\parallel =\parallel \chi \parallel \parallel h\parallel $, $t\in \mathbb{R}$. Hence we may apply the three lines theorem [47, Thm. 3.7] to the effect that $|f(z)|\le \parallel \chi \parallel \parallel h\parallel $ throughout the closed strip. Since χ and η are arbitrary, this entails $\parallel {\Delta}_{H}^{-iz}{E}_{H}\eta \parallel \le \parallel {E}_{H}\eta \parallel \le \parallel \eta \parallel $ and hence $\parallel {\Delta}_{H}^{\alpha}{E}_{\tilde{H}}\parallel \le \parallel {\Delta}_{H}^{\alpha}{E}_{H}\parallel \le 1$.

**Proposition 3.9.**Let $H\subset \mathcal{H}$ be a closed real subspace. Then there exists a conjugation Γ on $\mathcal{H}$ such that $\Gamma H=H$.

**Proof.**We split $\mathcal{H}$ as in (3.22), and have to construct a conjugation on each of the three summands. Since H has no components in the first summand, and the second (complex linear) summand is contained in H, we can pick arbitrary conjugations on the first two summands. In other words, it is sufficient to consider the case where H is standard.

**Lemma 3.10.**Let $\tilde{H}\subset H$ be a non-trivial half-sided modular inclusion of standard subspaces. Then there exists no conjugation Γ with $\Gamma \tilde{H}=\tilde{H}$ and $\Gamma H=H$.

**Proof.**Suppose Γ is a conjugation with $\Gamma H=H$ and $\Gamma \tilde{H}=\tilde{H}$. Then Γ commutes with both modular operators, ${\Delta}_{H}$ and ${\Delta}_{\tilde{H}}$. In view of the anti-linearity of Γ and ${\Gamma}^{2}=1$, this implies in particular that $\Gamma {\Delta}_{H}^{-it}{\Delta}_{\tilde{H}}^{it}\Gamma ={\Delta}_{H}^{it}{\Delta}_{\tilde{H}}^{-it}$ for all $t\in \mathbb{R}$.

#### 3.3. ${\ell}^{p}$-Conditions and Fermionic Second Quantization

**Theorem 3.11.**[29] Let $H\subset \mathcal{H}$ be a closed real subspace and ${X}_{1}$ a selfadjoint operator on $\mathcal{H}$, satisfying the same assumptions as in Thm. 3.2, with the exception of the norm bound $\parallel {X}_{1}{E}_{\pm}\parallel <1$, which is not assumed.

**Theorem 3.12.**Let $H\subset \mathcal{H}$ be a closed real subspace.

- (a)
- If ${X}_{1}{E}_{H}$ is ${\ell}_{\mathbb{R}}^{p}$ for some $0<p\le 1$, then ${\Xi}_{H,{X}_{1}}^{-}$ is p-nuclear, and ${\ell}^{q}$ for $q>p/(1-p)$.
- (b)
- If ${\Xi}_{H,{X}_{1}}^{-}$ is ${\ell}^{p}$ for some $p>0$, then ${X}_{1}{E}_{H}$ is ${\ell}_{\mathbb{R}}^{p}$.
- (c)
- ${\Xi}_{H,{X}_{1}}^{-}$ is ${\ell}^{p}$ for all $p>0$ if and only if ${X}_{1}{E}_{H}$ is ${\ell}_{\mathbb{R}}^{p}$ for all $p>0$.

**Proof.**$(a)$ The proof is similar to that of Thm. 3.3, but simpler because we do not need to control the norm of ${X}_{1}{E}_{H}$. We again pick a conjugation Γ on $\mathcal{H}$, and consider the doubled system $\underline{H}\subset \underline{\mathcal{H}}$, $\underline{X}$ (3.13), invariant under $\underline{\Gamma}$. Then, as in the Bose case, we can write $\underline{H}={\underline{\Gamma}}^{+}{\mathcal{K}}_{+}+{\underline{\Gamma}}^{-}{\mathcal{K}}_{-}$ with complex closed subspaces ${\mathcal{K}}_{\pm}\subset \underline{\mathcal{H}}$, and the corresponding operators ${X}_{1}{E}_{\pm}$ lie in ${\ell}^{p}(\underline{\mathcal{H}})$ because ${X}_{1}{E}_{H}\in {\ell}_{\mathbb{R}}^{p}(\mathcal{H})$. We again proceed to the joint upper bound $T\in {\ell}^{p}(\underline{\mathcal{H}})$, satisfying ${T}^{2}\ge {|{X}_{1}{E}_{\pm}|}^{2}$.

## 4. ${\ell}^{p}$-Properties of Laplace-Beltrami Operators

**Theorem 4.1.**For any $n\in \mathbb{N}$, ${A}^{n}$ is essentially self-adjoint on ${C}_{0}^{\infty}(\mathcal{C})$ in ${L}^{2}(\mathcal{C})$.

**Theorem 4.2.**Let d be the dimension of $\mathcal{C}$ and $k,l,n\in {\mathbb{N}}_{0}$.

- (a)
- If $n>\frac{3}{4}d+\frac{1}{2}(k+1)$, then ${K}_{-n}\in {C}^{k}({\mathcal{C}}^{\times 2})$.
- (b)
- If $n>\frac{3}{4}d+k+l+\frac{1}{2}$ and ${\chi}_{1},{\chi}_{2}\in {C}_{0}^{\infty}(\mathcal{C})$, then the operator ${A}^{k}{\chi}_{1}{A}^{-n}{\chi}_{2}{A}^{l}$ is Hilbert-Schmidt.

**Proof.**Let us write ${A}_{x}$, resp. ${A}_{y}$, for the differential operator A acting on the variables x, resp. y, and note that ${({A}_{x}+{A}_{y})}^{n}{K}_{-n}(x,y)={2}^{n}\delta (x,y)$. Because A is elliptic on $\mathcal{C}$, ${A}_{x}+{A}_{y}$ is elliptic on ${\mathcal{C}}^{\times 2}$ and we may use the calculus of Sobolev wave front sets ([52] Ch. VIII, especially Cor. 8.4.9 and 8.4.10) to see that $W{F}^{(s)}({K}_{-n})=W{F}^{(s-2n)}(\delta )$. The right-hand side is empty when $s-2n<-\frac{d}{2}$, so the left-hand side is empty when $s<2n-\frac{d}{2}$. When α is a multiindex in x and y with $|\alpha |\le k$, then we may choose $s=k+d+1$ to find $W{F}^{(s-k)}({\partial}^{\alpha}{K}_{-n})\subset W{F}^{(s)}({K}_{-n})=\xd8$ when $n>\frac{3}{4}d+\frac{1}{2}(k+1)$. Note that d is half the dimension of ${\mathcal{C}}^{\times 2}$, so ${\partial}^{\alpha}{K}_{-n}(x,y)$ is continuous by Sobolev’s Lemma (Corollary 6.4.9 in [52]) and hence ${K}_{-n}(x,y)$ is in ${C}^{k}({\mathcal{C}}^{\times 2})$. This proves the first item. For the second item we note that ${\chi}_{1}{A}^{-n}{\chi}_{2}$ is in ${C}^{2(k+l)}({\mathcal{C}}^{\times 2})$, so acting with the operators ${A}^{k}$ and ${A}^{l}$ we obtain an integral kernel K for ${A}^{k}{\chi}_{1}{A}^{-n}{\chi}_{2}{A}^{l}$ which is still in ${C}^{0}({\mathcal{C}}^{\times 2})$ and compactly supported. This means that $K\in {L}^{2}({\mathcal{C}}^{\times 2})$ and hence the operator is Hilbert-Schmidt. ☐

**Proposition 4.3.**Let $\alpha \in \mathbb{R}$ and let $\chi ,\tilde{\chi}\in {C}^{\infty}(\mathcal{C})$ such that χ is bounded, $\tilde{\chi}$ has compact support and the supports of χ and $\tilde{\chi}$ are separated by a distance $\delta >0$. Then we have in $\mathcal{B}({L}^{2}(\mathcal{C}))$ the bound

**Proof.**The smooth function $F(\lambda ):={({\lambda}^{2}+{m}^{2})}^{\alpha}$ defines a tempered distribution with Fourier transform $\widehat{F}(s)$. On $s>0$ the distribution $g(s):={(s/m)}^{\nu}\widehat{F}(s/m)$ with $\nu :=\alpha +\frac{1}{2}$ satisfies ${g}^{\u2033}(s)+{s}^{-1}{g}^{\prime}(s)-(1+{\nu}^{2}{s}^{-2})g(s)=0$, which is a modified Bessel equation. This means that $g(s)={C}_{1}{K}_{\nu}(s)$ for some constant ${C}_{1}$, where the modified Bessel function ${K}_{\nu}(s)$ satisfies ${K}_{\nu}(s)\le \sqrt{\frac{\pi}{2s}}{e}^{-s}\left(\right)open="("\; close=")">1+\frac{|4{\nu}^{2}-1|}{8s}$ ([54] 8.451) and hence

**Lemma 4.4.**Let $V\subset \mathcal{C}$ be an open region and let Ψ be a partial differential operator of order r with smooth coefficients supported in V. For any $R\in {\mathbb{N}}_{0}$ such that $2R\ge r$, there are ${\eta}_{1},\dots ,{\eta}_{R}\in {C}^{\infty}(\mathcal{C})$ supported in V such that ${\parallel \Psi f\parallel}^{2}\le {\sum}_{k=0}^{R}{\parallel {\eta}_{k}{A}^{k}f\parallel}^{2}$ for all $f\in {C}_{0}^{\infty}(\mathcal{C})$.

**Proof.**By taking complex conjugates of the coefficients of Ψ we obtain another partial differential operator $\overline{\Psi}$. Note that ${\parallel \Psi f\parallel}^{2}\le \langle f,Xf\rangle $, where $X={\Psi}^{*}\Psi +{\overline{\Psi}}^{*}\overline{\Psi}$ is a symmetric partial differential operator of even order $2r$ with real coefficients supported in V. We will show by induction over r that for any such operator X we can estimate

**Theorem 4.5.**Let $\alpha ,\beta ,\gamma \in \mathbb{R}$ and let $\chi ,\tilde{\chi}\in {C}_{0}^{\infty}(\mathcal{C})$ such that $\chi \equiv 1$ on a neighborhood of $\mathrm{supp}(\tilde{\chi})$. Then ${A}^{\beta}(1-\chi ){A}^{\alpha}\tilde{\chi}{A}^{\gamma}$ is bounded.

**Proof.**When $\beta =\gamma =0$ the result follows immediately from Proposition 4.3. Now let $V,\tilde{V}\subset \mathcal{C}$ be open subsets such that ${V}^{c}:=\mathcal{C}\backslash \overline{V}$ contains the support of $1-\chi $, $\tilde{V}$ that of $\tilde{\chi}$, and $\overline{{V}^{c}}$ and $\overline{\tilde{V}}$ are disjoint. Note that $\tilde{V}$ is relatively compact, so that $\overline{{V}^{c}}$ and $\overline{\tilde{V}}$ are separated by a minimal distance $\delta >0$.

**Lemma 4.6.**Let T be an operator in ${L}^{2}(\mathcal{C})$ defined on ${C}_{0}^{\infty}(\mathcal{C})$, let $\chi \in {C}_{0}^{\infty}(\mathcal{C})$ and assume that $T\chi {A}^{n}$ is bounded for all $n\in {\mathbb{N}}_{0}$. Then $T\chi {A}^{\beta}$ is ${\ell}^{p}$ for all $p>0$ and all $\beta \in \mathbb{R}$.

**Proof.**Let $\beta \in \mathbb{R}$ and $N\in {\mathbb{N}}_{0}$ be arbitrary, set ${\chi}_{1}:=\chi $ and choose ${\chi}_{2},\dots ,{\chi}_{2N+1}\in {C}_{0}^{\infty}(\mathcal{C})$ such that ${\chi}_{n+1}\equiv 1$ on $\mathrm{supp}({\chi}_{n})$ for $1\le n\le 2N$. We then have for all $1\le n\le 2N$ and $k,l\in {\mathbb{N}}_{0}$,

**Corollary 4.7.**Under the assumptions of Theorem 4.5, ${A}^{\beta}(1-\chi ){A}^{\alpha}\tilde{\chi}{A}^{\gamma}$ is ${\ell}^{p}$ for all $p>0$.

## 5. The Modular ${\ell}^{p}$-Condition for Free Scalar Fields

**Theorem 5.1.**Every quasi-free Hadamard state on the Weyl algebra of a real free scalar quantum field on a globally hyperbolic spacetime satisfies the modular ${\ell}^{p}$-condition.

#### 5.1. The Modular ${\ell}^{p}$-Condition and the Symplectic Form

**Proposition 5.2.**Let $d:=\frac{1-\Sigma}{1+\Sigma}(1-R)$ and let $h:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ be any continuous function, so that $h(d)$ is a self-adjoint operator. Then $X\u25cb\kappa :=\kappa \u25cbh(d)$ defines an operator X on a dense domain of $\mathcal{H}$, which is essentially self-adjoint with closure $h(\delta )$.

**Proof.**Since d vanishes on $R\mathcal{K}$ and δ on ${({\mathcal{H}}^{\prime})}^{\perp}$, it suffices to consider the summands $(1-R)\mathcal{K}$ and ${\mathcal{H}}^{\prime}$. This is tantamount to the special case $R=0$, which we now consider. The operator $\sqrt{1+\Sigma}$ commutes with $h(d)$ and maps the domain of $h(d)$ into a core for $h(d)$, so $\overline{X}={U}^{*}h(d)U$. It only remains to verify that ${U}^{*}dU=\delta $. For this we note that the range of $U\kappa $ is the domain of ${d}^{\frac{1}{2}}$ and the range of κ is a core for ${\delta}^{\frac{1}{2}}$. Furthermore, using the one-particle Tomita operator s we may compute for all ${f}_{1},{f}_{2}\in K+iK$:

**Proposition 5.3.**Define $c(\alpha ):={2}^{2\alpha}$ for $0<\alpha <\frac{1}{4}$ and $c(\alpha ):=1$ else. In the notations above we then have for all $p>0$ and $\alpha \in \mathbb{R}$:

- (a)
- $\parallel {\delta}^{\alpha}{|}_{\tilde{H}}{\parallel}_{\mathbb{R},p}=\parallel {\delta}^{\frac{1}{2}-\alpha}{{|}_{\tilde{H}}\parallel}_{\mathbb{R},p}$.
- (b)
- $\parallel {(1-{\Sigma}^{2})}^{\alpha}{|}_{\tilde{K}+i\tilde{K}}{\parallel}_{p}\le c(\alpha )\parallel {\delta}^{\alpha}{{|}_{\tilde{H}}\parallel}_{\mathbb{R},p}$.
- (c)
- If $\alpha \le \frac{1}{4}$, then $\parallel {\delta}^{\alpha}{|}_{\tilde{H}}{\parallel}_{\mathbb{R},p}\le {2}^{-2\alpha}c(\alpha )\parallel {(1-{\Sigma}^{2})}^{\alpha}{{|}_{\tilde{K}+i\tilde{K}}\parallel}_{p}$.

**Proof.**For any $\alpha \in \mathbb{R}$ and $f\in \tilde{K}$ in the domain of ${d}^{2\alpha}$ we have:

**Lemma 5.4.**Let X be a bounded positive operator on a complex Hilbert space $\mathcal{K}$, ${E}_{0}$ the projection onto its kernel and P any other projection.

- (a)
- If $n\in \mathbb{N}$ and $\alpha \ge {2}^{-n}$, then $\parallel {X}^{\alpha}{P\parallel}_{{2}^{n}p}\le {\parallel X\parallel}^{\alpha -{2}^{-n}}{\parallel PXP\parallel}_{p}^{{2}^{-n}}$.
- (b)
- If ${X}^{\alpha}P$ is ${\ell}^{p}$ for some $\alpha \le 0$ and $p>0$, then $\parallel (1-{E}_{0}){P\parallel}_{p}$ is finite too. In particular, if ${E}_{0}=0$, then P has a finite dimensional range. (Here ${X}^{\alpha}$ is defined with the convention of Proposition 5.3.)

**Proof.**By the polar decomposition, $\sqrt{X}P=U\sqrt{PXP}$ for some partial isometry U. Therefore, $\parallel P\sqrt{X}{P\parallel}_{2p}\le \parallel \sqrt{X}{P\parallel}_{2p}=\parallel \sqrt{PXP}{\parallel}_{2p}={\parallel PXP\parallel}_{p}^{\frac{1}{2}}$. By induction we find $\parallel P{X}^{{2}^{-n}}{P\parallel}_{{2}^{n}p}\le {\parallel PXP\parallel}_{p}^{{2}^{-n}}$ for all $n\in \mathbb{N}$, and hence $\parallel {X}^{{2}^{-n}}{P\parallel}_{{2}^{n}p}=\parallel P{X}^{{2}^{1-n}}{P\parallel}_{{2}^{n-1}p}^{\frac{1}{2}}\le {\parallel PXP\parallel}_{p}^{{2}^{-n}}$. When $\alpha \ge {2}^{-n}$ we may estimate $\parallel {X}^{\alpha}{P\parallel}_{{2}^{n}p}\le \parallel {X}^{\alpha -{2}^{-n}}\parallel \xb7\parallel {X}^{{2}^{-n}}{P\parallel}_{{2}^{n}p}\le {\parallel X\parallel}^{\alpha -{2}^{-n}}{\parallel PXP\parallel}_{p}^{{2}^{-n}}$.

**Corollary 5.5.**Let P denote the orthogonal projection in $\mathcal{K}$ onto $\tilde{K}+i\tilde{K}$, and $p>0$.

- (a)
- $\parallel P(1-{\Sigma}^{2}){P\parallel}_{p}^{\frac{1}{2}}\le \parallel {\delta}^{\frac{1}{4}}{{|}_{\tilde{H}}\parallel}_{\mathbb{R},2p}$.
- (b)
- If $\alpha \in (0,\frac{1}{4}]$, then $\parallel {\delta}^{\frac{1}{2}-\alpha}{|}_{\tilde{H}}{\parallel}_{\mathbb{R},p}=\parallel {\delta}^{\alpha}{|}_{\tilde{H}}{\parallel}_{\mathbb{R},p}\le {2}^{-2\alpha}c(\alpha ){\parallel P(1-{\Sigma}^{2})P\parallel}_{{2}^{-n}p}^{{2}^{-n}}$ for all $n\in \mathbb{N}$ such that $\alpha \ge {2}^{-n}$.
- (c)
- If ${\delta}^{\alpha}{|}_{\tilde{H}}$ is ${\ell}_{\mathbb{R}}^{p}$ for some $\alpha \notin (0,\frac{1}{2})$ and $p>0$, then ${\parallel (1-R)P\parallel}_{p}$ is finite. In particular, if $R=0$, then $\tilde{K}$ is finite dimensional.

**Proof.**This follows from Proposition 5.3 and Lemma 5.4 with $X=1-{\Sigma}^{2}$. For the first item we use the estimate

#### 5.2. Modular ${\ell}^{p}$-Condition for Quasi-Free Hadamard States

**Proposition 5.6.**For any regular Cauchy pair $(\tilde{V},V)$ in $\mathcal{C}$ and any $p>0$ the operator ${P}_{{\mu}_{0},\tilde{V}}(1-{\Sigma}_{{\mu}_{0},V}^{2}){P}_{{\mu}_{0},\tilde{V}}$ is ${\ell}^{p}$.

**Proof.**We let $U:{\mathcal{K}}_{{\mu}_{0}}\to {L}^{2}{(\mathcal{C})}^{\oplus 2}$ be the unitary map defined by $U(\phi ,\pi ):=\frac{1}{\sqrt{2}}({A}^{\frac{1}{4}}\phi ,{A}^{-\frac{1}{4}}\pi )$ and we choose $\chi ,\tilde{\chi}\in {C}_{0}^{\infty}(V)$ such that $\tilde{\chi}\equiv 1$ on $\tilde{V}$ and $\chi \equiv 1$ on a neighborhood of $\mathrm{supp}(\tilde{\chi})$. We then define operators X and $\tilde{X}$ on $\mathcal{D}(\mathcal{C})$ by $\tilde{X}(\phi ,\pi ):=(\tilde{\chi}\phi ,\tilde{\chi}\pi )$ and $X(\phi ,\pi ):=(\chi \phi ,\chi \pi )$. These operators are closable and we denote their closures by the same symbol. For any $f\in \mathcal{D}(\tilde{V})$ we have $\tilde{X}f=f$ and hence ${P}_{{\mu}_{0},\tilde{V}}=\tilde{X}{P}_{{\mu}_{0},\tilde{V}}$. Similarly, ${P}_{{\mu}_{0},V}X=X$, which implies $1-{P}_{{\mu}_{0},V}=(1-{P}_{{\mu}_{0},V})(1-X)$. (Note how the ordering of these products matches the chosen support properties of $\tilde{\chi}$ and χ.) Using ${P}_{{\mu}_{0},\tilde{V}}{P}_{{\mu}_{0},V}={P}_{{\mu}_{0},\tilde{V}}$ and ${\Sigma}^{2}=1$ (5.3) we then have

**Lemma 5.7.**Let $(\tilde{V},V)$ be a regular Cauchy pair in $\mathcal{C}$ and assume that ${\omega}_{\mu}$ and ${\omega}^{0}$ are quasi-equivalent states on $\mathcal{W}(D(V))$. Then, for any $p>0$:

**Proof.**The map ${U}_{V}f:=\sqrt{{M}_{V}}f$ defines a unitary isomorphism from ${\mathcal{K}}_{\mu ,V}$ to ${\mathcal{K}}_{{\mu}_{0},V}$ and we have ${U}_{V}{\Sigma}_{\mu ,V}{U}_{V}^{*}={M}_{V}^{-\frac{1}{2}}{\Sigma}_{{\mu}_{0},V}{M}_{V}^{-\frac{1}{2}}$, because ${\Sigma}_{{\mu}_{0},V}={M}_{V}{\Sigma}_{\mu ,V}$ on $\mathcal{D}(V)$. Moreover, for any $\tilde{V}\subset V$, ${U}_{V}{P}_{\mu ,\tilde{V}}{U}_{V}^{*}$ is the orthogonal projection onto the range of $\sqrt{{M}_{V}}{P}_{{\mu}_{0},\tilde{V}}$, which means in particular that

**Theorem 5.8.**For any relatively compact open region $V\subset \mathcal{C}$ and any quasi-free Hadamard state ${\omega}_{\mu}$, the operator ${M}_{V}-1$ is ${\ell}^{p}$ for all $p>0$.

**Proof.**This follows essentially from Proposition 3.8 of [60] and its proof, together with the following comments. [60] uses a spacetime formulation for the proof of its Proposition 3.8, but this is unitarily equivalent to the initial value formulation we use here. For any relatively compact region $W\subset \mathcal{C}$ containing V, the proof of Proposition 3.8 in [60] proves the existence of sequences of real elements ${F}_{k},{F}_{k}^{\prime}\in \mathcal{D}(W)$ such that

**Proof.**We first consider a quasi-free Hadamard state ${\omega}_{\mu}$ in a standard ultra-static spacetime. From Theorem 5.8, Lemma 5.7 and Proposition 5.6 we see that ${P}_{\mu ,\tilde{V}}(1-{\Sigma}_{\mu ,V}^{2}){P}_{\mu ,\tilde{V}}$ is ${\ell}^{p}$ for all $p>0$ and all regular Cauchy pairs $(\tilde{V},V)$ in $\mathcal{C}$. By Proposition 5.3 and Corollary 5.5 this means that ${\delta}_{\mu ,V}^{\alpha}{|}_{{H}_{\tilde{V}}}$ is ${\ell}^{p}$ for all $p>0$.

#### 5.3. On Non-Hadamard States Satisfying the Modular ${\ell}^{p}$-Condition

**Example:**Our example of a non-Hadamard state with the modular ${\ell}^{p}$-property is a quasi-free state ${\omega}_{\mu}$ for the massive free scalar field on a standard ultra-static spacetime. In fact, we will choose ${\omega}_{\mu}$ to be quasi-equivalent to the ground state ${\omega}^{0}$, given by ${\mu}_{0}$. To define μ, we choose an arbitrary vector $\psi \in {\mathcal{K}}_{{\mu}_{0},\mathcal{C}}$ which is given by real initial data on $\mathcal{C}$, $\Gamma \psi =\psi $. Let ${P}_{\psi}$ be the orthogonal projection onto the linear space spanned by ψ and set ${M}_{\mathcal{C}}:=I+{P}_{\psi}$. We note that $\Gamma {M}_{\mathcal{C}}\Gamma ={M}_{\mathcal{C}}$ and we define $\mu ({f}_{1},{f}_{2}):={\mu}_{0}({f}_{1},{M}_{\mathcal{C}}{f}_{2})$. This defines a real inner product on $\mathcal{D}(\mathcal{C})$ which satisfies $\mu (f,f)\ge {\mu}_{0}(f,f)$ and therefore the bound (5.1), so it defines a quasi-free state ${\omega}_{\mu}$.

## 6. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Modular Nuclearity: A Generally Covariant Perspective. *Axioms* **2016**, *5*, 5.
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