October 30, 2025
This note presents a canonical construction of global observables -sometimes referred to in the literature as macroscopic observables or observables at infinity- in statistical mechanics, providing a unified treatment of both commutative and non-commutative cases. Unlike standard approaches, the framework is formulated directly in the \(C^*\)-algebraic setting, without relying on any specific representation.
In the statistical mechanics of lattice systems, one typically studies large collections of interacting particles arranged on a regular lattice, and the behavior of these systems in appropriate limiting regimes reveals their macroscopic properties. Two distinct types of such “infinite volume” limits are commonly considered.
The first is the thermodynamic limit, in which the lattice size tends to infinity while local observables are examined within finite regions of the lattice. This limit is described by the so-called quasi-local algebra and allows one to rigorously define expectation values of local quantities and ensures that bulk properties, such as energy density or correlation functions, stabilize in the infinite-volume system. Its significance is well established in the context of phase transitions and equilibrium phenomena in quantum and classical statistical mechanics [1]–[3].
The second, less standard but equally important, is what is sometime referred to as macroscopic limit [4], [5], which focuses on observables that are often called “global” or “observables at infinity,” such as spatial averages of local quantities over regions of diverging size [6]–[8]. Unlike the thermodynamic limit, which addresses the stability of local observables under volume growth, the macroscopic limit captures collective, non-local behavior that reflects emergent classical features of quantum systems [9], [10]. While this construction is classical in spirit and well established in statistical mechanics, its quantum counterpart is considerably less developed, often requiring sophisticated representation-theoretic methods [1], [2].
In this work we develop a \(C^*\)-algebraic framework to study the macroscopic behavior of statistical mechanical systems, providing a unified treatment of both commutative and non-commutative settings. This approach emphasizes the connection between quantum and classical systems and does not require the use of any particular representation. In the non-commutative setting, the resulting \(C^*\)-algebra naturally contains commutative \(C^*-\) subalgebras generated by macroscopic averages, thereby providing a bridge between classical thermodynamics as a limit of quantum statistical mechanics. In the commutative case, it is proved that the ensuing \(C^*-\)algebra can be identified with the well-known algebra of \(\sigma\)-tail measurable functions, underscoring the significance of this construction.
Let \(\Gamma \subset \mathbb{R}^d\) be a countable set. This already endows the set of finite subsets of \(\Gamma\) with a partial order (inclusion), which is upward directed and hence
defines a notion of convergence, namely “convergence along the net of finite subsets of \(\Gamma\), directed by inclusion”, that is \[\lim_{\Lambda \nearrow \Gamma} F(\Lambda) = \alpha\]
means that for every \(\varepsilon > 0\), there exists a finite subset \(K_\epsilon \subset \Gamma\) such that \[\|F(\Lambda) - \alpha\| < \varepsilon \quad
\text{whenever } K_\epsilon \subset \Lambda.\] Here, \(F\) is a function (or operator) defined on finite subsets of \(\Gamma\) and taking values in a normed space.1
To each \(x \in \Gamma\) we associate a unital \(C^*\)-algebra \(\mathcal{B}_x\); in fact, we assume the same algebra \(\mathcal{B}\) for all \(x\), and use \(x\) only to denote the lattice position. The (minimal) tensor product of \(\otimes_{x\in\Lambda}\mathcal{B}_x\) is denoted by \(\mathcal{B}_\Lambda\) and ensures that \(\mathcal{B}_\Lambda\) is again a \(C^*\)-algebra. We then consider the full \(C^*\)-product \(\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\) defined by
\[\begin{align}
\label{Eq:32full32C42-product}
\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda:=\{(a_\Lambda)_\Lambda\,|\,(\|a_\Lambda \|_\Lambda)_\Lambda\in\ell^\infty(\Gamma)\}\,,
\end{align}\tag{1}\] where \((a_\Lambda)_\Lambda\) should be understood as element in the algebraic direct product with pointwise operations. As it is well-known [11] \(\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\) is a \(C^*\)-algebra with respect to sup
norm \(\|(a_\Lambda)_\Lambda\|_\infty:=\sup\limits_{\Lambda\Subset\Gamma}\|a_\Lambda\|_\Lambda\). For \((a_\Lambda)_\Lambda,(b_\Lambda)_\Lambda\in
\Pi_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\), we introduce the following \(\sim\)-equivalence relation \[\begin{align}
\label{Eq:32equivalence32relation32among32sequences}
(a_\Lambda)_\Lambda\sim(b_\Lambda)_\Lambda\Longleftrightarrow \lim_{\Lambda\nearrow\Gamma}\|a_\Lambda-b_\Lambda\|_\Lambda=0\,.
\end{align}\tag{2}\] For each given sequence \((a_\Lambda)_\Lambda\), we will denote by \([a_\Lambda]_\Lambda:=[(a_\Lambda)_\Lambda]\) the corresponding equivalence class with
respect to 2 . Moreover, the direct \(C^*\)-sum \[\begin{align}
\label{Eq:32direct32C42-sum}
\bigoplus_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda:&=\{(a_\Lambda)_\Lambda\in\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\,|\,
(\|a_\Lambda\|_\Lambda )_\Lambda\in C_0(\Gamma)\}\\&=\{(a_\Lambda)_\Lambda\in\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\,|\,
\lim_{\Lambda\nearrow\Gamma}\|a_\Lambda\|_\Lambda=0\}\,
\end{align}\tag{3}\] is a closed two-sided ideal in \(\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\) and thus we may consider the quotient \[\begin{align}
\label{Eq:32quotient32algebra}
[\mathcal{B}]_\sim:=\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda/\bigoplus_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\,,
\end{align}\tag{4}\] which is nothing but the space of \(\sim\)-equivalence classes \([a_\Lambda]_\Lambda\) for bounded sequences \((a_\Lambda)_\Lambda\), i.e. \[[\mathcal{B}]_\sim=q(\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda)\] where \(q\) is the canonical quotient map, \[q : \prod_{\Lambda\Subset\Gamma} \mathcal{B}_\Lambda \;\longrightarrow\; \prod_{\Lambda\Subset\Gamma} \mathcal{B}_\Lambda/\bigoplus_{\Lambda\Subset\Gamma} \mathcal{B}_\Lambda, \quad
q\big( (a_\Lambda)_{\Lambda\Subset\Gamma} \big) = [ (a_\Lambda)_{\Lambda \Subset \Gamma} ]_\Lambda,\] Importantly, \([\mathcal{B}]_\sim\) is a \(C^*\)-algebra with norm
\[\begin{align}
\label{Eq:32norm32of32the32quotient32algebra} \|[a_\Lambda]_\Lambda\|_{[\mathcal{B}]_\sim}=\limsup_{\Lambda\nearrow \Gamma}\|a_\Lambda\|_\Lambda\,.
\end{align}\tag{5}\] Hence, passing to the quotient makes it possible to identify sequences that represent the same observable in the limit as \(\Lambda\nearrow\Gamma\), so that only essentially distinct
observables are captured. Of course, \([\mathcal{B}]_\sim\) is very large, so that one typically focuses on suitable \(C^*\)-sub algebras.
Example 1 (Quasi-local algebra). We consider the \(*\)-algebra of local observables \[\dot{\mathcal{B}}^\infty:= \bigcup_{\Lambda\Subset\Gamma}
\mathcal{B}_\Lambda.\] Here, we implicitly (injectively) embedded \(\mathcal{B}_\Lambda\) into the full \(C^*\)- product \[\prod_{\Lambda\Subset\Gamma}
\mathcal{B}_\Lambda\] by identifying an element \(b_{\Lambda'} \in \mathcal{B}_{\Lambda'}\) with a local sequence \((b_\Lambda)_{\Lambda\Subset \Gamma}\) defined as \[b_\Lambda := \begin{cases}
b_{\Lambda'} \otimes \mathbf{1}_{\Lambda\setminus\Lambda'}, & \Lambda' \subset \Lambda, \\
0, & \text{otherwise},
\end{cases}\] where \(\mathbf{1}_{\Lambda\setminus\Lambda'}\) denotes the identity in \(\mathcal{B}_{\Lambda\setminus\Lambda'}=\otimes_{x\in
{\Lambda\setminus\Lambda'}}\mathcal{B}_x\).
The algebra \(\dot{\mathcal{B}}^\infty\) is a sub-algebra of \(\prod_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda\). The quasi-local algebra** is then defined to be the completion of
the quotient \[[\mathcal{B}]^\infty:=\overline{q(\dot{\mathcal{B}}^\infty)}^{\|\cdot\|}\subset [\mathcal{B}]_\sim,\] where norm is given by \(\|[a_\Lambda]_\Lambda\|_{[\mathcal{B}]^\infty}=\limsup_{\Lambda\nearrow\Gamma}\|a_\Lambda\|_\Lambda\), which can be shown to equal the actual limit. It is not difficult to see that \([\mathcal{B}]^\infty\) is a \(C^*\)-subalgebra of \([\mathcal{B}]_\sim\). The quasi-local algebra provides the standard framework for describing the
thermodynamic limit in statistical mechanics [1], [2]. \(\blacksquare\)**
The \(C^*\)-algebra \([\mathcal{B}]_\sim\) contains many other elements for which the expectation value is not defined in several physically relevant states, such as translation-invariant states. We give some examples of such sequences.
Example 2. We consider the following sequences of tensor products of matrices:
\[a_\Lambda := \bigotimes_{x \in \Lambda} \sigma^1_x,\] where \(\sigma^1_x\) is the \(1\)-component of the Pauli matrix at site \(x\).
\[a_\Lambda := \bigotimes_{x \in \Lambda} \begin{cases} \sigma^1_x, & x \text{ odd},\\ \sigma^3_x, & x \text{ even}. \end{cases}\]
Define a sequence \((a_\Lambda)_\Lambda\) by partitioning \(\Gamma\) into consecutive blocks of strictly increasing lengths \((B_n)_{n\ge0}\), assigning \(\sigma^x\) to even-indexed blocks and \(\sigma^z\) to odd-indexed blocks, and setting \[a_\Lambda := \bigotimes_{x \in \Lambda} A_x, \quad A_x := \begin{cases} \sigma^x & \text{if x is in an even-indexed block,}\\ \sigma^z & \text{if x is in an odd-indexed block.} \end{cases}\]
For a translation-invariant state \(\omega\), the limit \[\lim_{\Lambda \nearrow \Gamma} \omega(a_\Lambda)\] is generally not defined for these sequences, because the product over infinitely many fluctuating operators diverges or oscillates.
We now construct another \(C^*\)-sub-algebra of \([\mathcal{B}]_\sim\). Let \[\mathcal{C}^\infty:=\left\{ (a_\Lambda)_\Lambda \in \prod_{\Lambda\Subset\Gamma} \mathcal{B}_\Lambda : \forall \quad (b_\Lambda)_\Lambda \in \dot{ \mathcal{B}}^\infty, \;\lim_{\Lambda \nearrow \Gamma} \|[a_\Lambda, b_\Lambda]\|_\Lambda = 0 \right\}.\]
Lemma 1. \(\mathcal{C}^\infty\) is a norm-closed *-subalgebra of \(\prod_\Lambda \mathcal{B}_\Lambda\), and hence a \(C^*\)-algebra.
Proof. We must prove the following three conditions.
\(*\)-algebra property: Let \(a=(a_\Lambda)_\Lambda, a'=(a_\Lambda')_{\Lambda}\in \mathcal{C}^\infty\) and \(b=(b_\Lambda)_\Lambda\in \dot{\mathcal{B}}^\infty\). Then \[[a+a',b] = [a,b] + [a',b], \quad [a^*,b] = [a,b^*]^*, \quad [aa',b] = a[a',b] + [a,b]a'.\] Since \(\|[a_\Lambda,b_\Lambda]\|\to 0\) and \(\|[a'_\Lambda,b_\Lambda]\|\to 0\), the same holds for \(a+a'\), \(a^*\), and \(aa'\). Thus \(\mathcal{C}^\infty\) is a *-subalgebra.
Norm-closedness: Let \((a^{(n)})\subset \mathcal{C}^\infty\) be a net that converges in the sup-norm to \(a\), i.e. \(\|a^{(n)}-a\|_\infty \to 0\). Then for any \(b\in \dot{\mathcal{B}}^\infty\) and each \(\Lambda\), \[\|[a_\Lambda,b_\Lambda]\| \le \|[a_\Lambda - a^{(n)}_\Lambda, b_\Lambda]\| + \|[a^{(n)}_\Lambda,b_\Lambda]\| \le 2 \|a - a^{(n)}\|_\infty \|b_\Lambda\| + \|[a^{(n)}_\Lambda,b_\Lambda]\|.\] Taking the limit \(\Lambda\nearrow \Gamma\) and then \(n\to\infty\), the right-hand side goes to \(0\). Hence \(a\in \mathcal{C}^\infty\).
\(C^*\)-property: Being a closed *-subalgebra of the C*-algebra \(\prod_\Lambda \mathcal{B}_\Lambda\) with the sup-norm, \(\mathcal{C}^\infty\) is itself a \(C^*\)-algebra.
◻
Consider the quotient \[[\mathcal{C}]^\infty:=\mathcal{C}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{B}_\Lambda =q(\mathcal{C}^\infty),\] where \(q\) is the canonical projection onto \([\mathcal{B}]_\sim\). It follows that \([\mathcal{C}]^\infty\) is a \(C^*\)-algebra, and \[[\mathcal{C}]^\infty=[\mathcal{B}]_\sim\cap (q(\dot{\mathcal{B}}^\infty))',\] is the relative commutant of the \(*\)-algebra \([\dot{\mathcal{B}}]^\infty\) in the \(C^*\)-algebra \([\mathcal{B}]_\sim\), and therefore norm-closed. Furthermore, we may define \[\mathcal{C}_\Lambda^\infty:=\left\{ (a_{\Lambda'})_{\Lambda'} \in \prod_{\Lambda'\Subset\Gamma} \mathcal{B}_{\Lambda'} : \forall \quad b_\Lambda\in \mathcal{B}_\Lambda, \;\lim_{\Lambda' \nearrow \Gamma} \|[a_{\Lambda'}, b_\Lambda]\|_{\Lambda'} = 0 \right\},\] where \(b_\Lambda\) has to be understood as element of \(\mathcal{B}_{\Lambda'}\) under the canonical embedding. It then follows \[\mathcal{C}^\infty=\bigcap_{\Lambda \Subset \Gamma} \mathcal{C}_\Lambda^\infty.\] Motivated by the pioneering ideas described in [6], we introduce the following definition.
Definition 1. The \(C^*\)-subalgebra \([\mathcal{C}]^\infty\) is called the algebra of global observables. \(\blacksquare\)
The \(C^*\)-algebra \([\mathcal{C}]^\infty\) turns out to be non-commutative.
Lemma 2. \([\mathcal{C}]^\infty\) is non-commutative.
Proof. Consider the sequences \((a_\Lambda)_\Lambda\) and \((c_\Lambda)_\Lambda\), defined by \[a_\Lambda := \mathbf{1}_{\Lambda \setminus
\{x_\Lambda\}} \otimes a_{x_\Lambda}, \quad
c_\Lambda := \mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes c_{x_\Lambda},\] where \(a_{x_\Lambda}, c_{x_\Lambda}\) are fixed non-commuting observables acting on site \(x_\Lambda \in
\Lambda\), i.e., local observables in \(\mathcal{B}_{\{x_\Lambda\}}\), such that \([a_{x_\Lambda}, c_{x_\Lambda}] \neq 0\). These sequences are what we call “local sequences
translated to infinity,” where \(x_\Lambda\) is chosen so that \(x_\Lambda \to \infty\) as \(\Lambda \nearrow \Gamma\).
For any local observable \(b \in \dot{\mathcal{B}}^\infty\), supported on a finite subset \(\Lambda' \Subset \Gamma\), define the sequence \[b_\Lambda :=
\begin{cases}
b_{\Lambda'} \otimes \mathbf{1}_{\Lambda \setminus \Lambda'}, & \Lambda' \subseteq \Lambda, \\
0, & \text{otherwise}.
\end{cases}\] Since \(a_\Lambda\) acts non-trivially only on site \(x_\Lambda\) which eventually is supported outside \(\Lambda'\) and \(b_\Lambda\) acts non-trivially only on sites in \(\Lambda'\), their supports are disjoint for all sufficiently large \(\Lambda\). Hence, \[[a_\Lambda, b_\Lambda] = 0 \quad \text{for all sufficiently large } \Lambda.\] Therefore, \[\lim_{\Lambda \nearrow \Gamma} \|[a_\Lambda, b_\Lambda]\| = 0,\] and similarly for \(c_\Lambda\). This shows \[(a_\Lambda)_\Lambda, (c_\Lambda)_\Lambda \in \dot{\mathcal{C}}^\infty.\] By definition, \[[a_\Lambda, c_\Lambda]
= (\mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes a_{x_\Lambda})(\mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes c_{x_\Lambda})
- (\mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes c_{x_\Lambda})(\mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes a_{x_\Lambda}).\] Using standard tensor product properties, we find \[[a_\Lambda, c_\Lambda] =
\mathbf{1}_{\Lambda \setminus \{x_\Lambda\}} \otimes [a_{x_\Lambda}, c_{x_\Lambda}].\] Applying the operator norm, we obtain \[\|[a_\Lambda, c_\Lambda]\|_\Lambda = \|[a_{x_\Lambda}, c_{x_\Lambda}]\|,\] where the
norm on the right-hand side does not depend on \(\Lambda\). Recall that the norm on the quotient algebra \([\mathcal{C}]^\infty\) is defined as \[\|[a,
c]\|_{[\mathcal{C}]^\infty} := \limsup_{\Lambda \nearrow \Gamma} \|[a_\Lambda, c_\Lambda]\|.\] Since \(\|[a_\Lambda, c_\Lambda]\|_\Lambda = \|[a_{x_\Lambda}, c_{x_\Lambda}]\|\) for all \(\Lambda\), and the commutator \([a_{x_\Lambda}, c_{x_\Lambda}] \neq 0\), we conclude that \[\|[a, c]\|_{[\mathcal{C}]^\infty} > 0.\] Therefore, the elements
\(a = [a_\Lambda]_\Lambda\) and \(c = [c_\Lambda]_\Lambda\) in \([\mathcal{C}]^\infty\) do not commute. As a result, the algebra \([\mathcal{C}]^\infty\) is non-commutative. ◻
Besides the observables “translated to infinity” considered in Lemma 2, we give another example of a global observable.
Example 3. Set \(\Gamma=\mathbb{Z}\). Consider the following sequence of local tensor products: \[c_N := \bigotimes_{k=1}^{N - \lfloor N/2 \rfloor} 1 \;\otimes\; \bigotimes_{j=1}^{\lfloor N/2 \rfloor} a,\] where \(1\) denotes the identity operator on a single site and \(a\) is a non-trivial and non-zero single-site matrix with \(\|a\|\le 1\). This sequence looks like \[\begin{align} c_1 &= 1, \\ c_2 &= 1 \otimes a, \\ c_3 &= 1 \otimes 1 \otimes a, \\ c_4 &= 1 \otimes 1 \otimes a \otimes a, \\ c_5 &= 1 \otimes 1 \otimes 1 \otimes a \otimes a, \\ c_6 &= 1 \otimes 1 \otimes 1 \otimes a \otimes a \otimes a, \\ c_7 &= 1 \otimes 1 \otimes 1 \otimes 1 \otimes a \otimes a \otimes a, \\ c_8 &= 1 \otimes 1 \otimes 1 \otimes 1 \otimes a \otimes a \otimes a \otimes a. \end{align}\] Note that the sequence \((c_N)_N\) is uniformly bounded, and for any fixed local observable \(b\), \(c_N\) asymptotically commutes with \(b\). Therefore, \((c_N)_N\) defines a legitimate element in \([\mathcal{C}]^\infty\). However \((c_N)_N\) does in general not commute with the sequences constructed in the proof Lemma 2. \(\blacksquare\)
In the next section, we will prove that \([\mathcal{C}]^\infty\) admits an interesting commutative \(C^*\)-subalgebra, namely the one generated by “macroscopic averages”.
We construct a \(C^*\)-subalgebra of \([\mathcal{C}]^\infty\), containing all “macroscopic averages". This algebra enjoys the property of being commutative, and therefore resembles a
classical observable algebra describing the macroscopic limit arising from the underlying quantum statistical mechanics.
To illustrate this idea, we focus on the one-dimensional lattice i.e., \(\Gamma = \mathbb{Z}\). Let \(\Lambda_1\subset \Lambda_2\subset\dots\) be a strictly increasing sequence of connected
finite subsets of \(\mathbb{Z}\), with \(|\Lambda_N|=N\) and \(\cup_N\Lambda_N=\mathbb{Z}\). For each region \(\Lambda=\Lambda_N\), we consider the linear operator (left-shift operator) \[\gamma_\Lambda \colon \mathcal{B}_\Lambda \to \mathcal{B}_\Lambda,\] uniquely defined by continuous and linear
extension of the following map on elementary tensors: \[\begin{align}
\label{Eq:32gamma-operator} \gamma_{\Lambda}(a_{1} \otimes \ldots \otimes a_{N}) := a_{2} \otimes \ldots \otimes a_{N} \otimes a_{1}\,,
\end{align}\tag{6}\] where \(a_{1}, \dots, a_{N} \in \mathcal{B}\). The operator \(\gamma_\Lambda\) is a \(*\)-endomorphism of the algebra
\(\mathcal{B}_\Lambda\). Moreover, \(\gamma_\Lambda^{N} = \operatorname{id}\). We then define the averaged shift operator
\[\begin{align}
\label{Eq:32averaged32gamma-operator} \overline{\gamma}_\Lambda := \frac{1}{N} \sum_{j=0}^{N-1} \gamma_\Lambda^j\,.
\end{align}\tag{7}\] The image of this operator, \[\begin{align}
\label{Eq:32gamma32invariant32subalgebra} \mathcal{B}^\gamma_\Lambda := \overline{\gamma}_\Lambda(\mathcal{B}_\Lambda),
\end{align}\tag{8}\] is a \(C^*\)-subalgebra of \(\mathcal{B}_\Lambda\) consisting of the \(\gamma\)-invariant elements.
Definition 2. A sequence \((a_\Lambda)_\Lambda\) is called a \(\gamma\)-sequence if there exists a finite subset \(\Lambda_0 \Subset \Gamma\) and an element \(a_{\Lambda_0} \in \mathcal{B}_{\Lambda_0}\) such that \[\begin{align} \label{Eq:32gamma-sequence32property} a_\Lambda = \overline{\gamma}_\Lambda^{\Lambda_0}(a_{\Lambda_0}) := \begin{cases} \overline{\gamma}_\Lambda\left( \mathbf{1}_{\Lambda \setminus \Lambda_0} \otimes a_{\Lambda_0} \right), & \Lambda \supseteq \Lambda_0, \\ 0, & \text{otherwise}, \end{cases} \end{align}\qquad{(1)}\] where \(\mathbf{1}_{\Lambda \setminus \Lambda_0}\) denotes the identity on the complementary tensor factors. \(\blacksquare\)
In what follows, we consider the \(*\)-algebra \(\dot{\mathcal{B}}^\infty_\gamma \subset \prod_{\Lambda \Subset \Gamma} \mathcal{B}_\Lambda\) generated by all \(\gamma\)-sequences, together with its image under the canonical projection \([\dot{\mathcal{B}}]^\infty_\gamma \subset [\mathcal{B}]_\sim^\infty\). The algebra \([\dot{\mathcal{B}}]^\infty_\gamma\) enjoys remarkable properties; in particular, it can be completed to a commutative \(C^*\)-algebra \([\mathcal{B}]^\infty_\gamma\) [9]. Moreover, the \(\gamma\)-sequences constitute a continuous field of \(C^*\)-algebras over \(\mathbb{N}\cup\{\infty\}\) with limit \([\mathcal{B}]^\infty_\gamma\).
We begin with the following observation, which relates states on \([\mathcal{B}]_\gamma^\infty\) to translationally invariant states on the quasi-local algebra.
Remark 1. Each state on \([\mathcal{B}]^\infty_\gamma\) canonically induces a translationally invariant state on the quasi-local algebra \([\mathcal{B}]^\infty\).
Indeed, one can prove that the map \[\begin{align} &\overline{\gamma}_\infty: [\mathcal{B}]^\infty\to [\mathcal{B}]^\infty_\gamma;\\ &[a_\Lambda]_\Lambda\mapsto [\overline{\gamma}_\Lambda(a_{\Lambda})]_\Lambda
\end{align}\] is well-defined. In particular, applying a state \(\omega\in S([\mathcal{B}]^\infty_\gamma)\) to an equivalence class of \(\gamma\)-sequences, yields \[\omega([\overline{\gamma}_\Lambda(a_{\Lambda})])=\omega\circ \overline{\gamma}_\infty([a_\Lambda]_\Lambda)\] If now we define \[\hat{\omega}:=\omega\circ \overline{\gamma}_\infty,\] then \(\hat{\omega}\) is a translationally invariant state on \([\mathcal{B}]^\infty\), since for any \(j\) \[\hat{\omega}\circ
\gamma_\infty^j=\hat{\omega},\] where \[\gamma_\infty^j([a_\Lambda]_\Lambda)=[\gamma_\Lambda^j(a_\Lambda)]_\Lambda\] is the translation of \(j\) lattice sites.
In this way, the expectation values of \(\gamma\)-sequences in \(\hat{\omega}\) capture the macroscopic averages of local observables and thereby encode the distribution over ergodic
components of the corresponding translation-invariant state, see [1] for further details hereon. \(\blacksquare\)
More importantly, \([\mathcal{B}]^\infty_\gamma\) is a \(C^*\)-subalgebra of \([\mathcal{C}]^\infty\).
Proposition 2. \([\mathcal{B}]^\infty_\gamma\subset [\mathcal{C}]^\infty\).
Proof. The assertion follows directly from the construction. We first prove it for \(\gamma\)-sequences; the general case then follows immediately from [9]. For any \(\gamma\)-sequence \((a_\Lambda)_\Lambda\) as in Definition 2, we claim that for all local observables \((b_\Lambda)_\Lambda \in \dot{\mathcal{B}}^\infty\) \[\lim_{\Lambda \nearrow \Gamma} \|[a_\Lambda, b_\Lambda]\|_\Lambda = 0.\] To see this, fix \(\Lambda_0 \Subset \Gamma\) and \(a_{\Lambda_0} \in \mathcal{B}_{\Lambda_0}\) such that \[a_\Lambda = \overline{\gamma}_{\Lambda}^{\Lambda_0}(a_{\Lambda_0}) = \frac{1}{N} \sum_{j=0}^{N-1} \gamma_\Lambda^j \big( 1_{\Lambda \setminus \Lambda_0} \otimes a_{\Lambda_0} \big), \qquad \Lambda \supseteq \Lambda_0,\] and \(a_\Lambda = 0\) otherwise. Let \((b_\Lambda)_\Lambda\) be a local observable supported on some fixed \(\Lambda' \Subset \Gamma\). Since \(b_\Lambda\) acts non-trivially only on the tensor factors associated with \(\Lambda'\), and \(a_\Lambda\) is an average over cyclic shifts, the commutator norm can be estimated as \[\begin{align} \| [a_\Lambda, b_\Lambda] \|_\Lambda &= \left\| \frac{1}{N} \sum_{j=0}^{N-1} \left[ \gamma_\Lambda^j \big( 1_{\Lambda \setminus \Lambda_0} \otimes a_{\Lambda_0} \big), b_\Lambda \right] \right\|_\Lambda \\ &\leq \frac{1}{N} \sum_{j=0}^{N-1} \left\| \left[ \gamma_\Lambda^j \big( 1_{\Lambda \setminus \Lambda_0} \otimes a_{\Lambda_0} \big), b_\Lambda \right] \right\|_\Lambda. \end{align}\]
For large \(\Lambda\), most of the shifts \(\gamma_\Lambda^j\) move the support of \(a_{\Lambda_0}\) away from the support of \(b_{\Lambda}\). More precisely, at most \(|\Lambda_0| + |\Lambda'|\) terms in the sum correspond to non-disjoint supports, and hence potentially non-zero commutators. For all other terms, the supports are disjoint, so the commutator vanishes. Thus, \[\| [a_\Lambda, b_\Lambda] \|_\Lambda \leq \frac{|\Lambda_0|+|\Lambda'|}{N}\; 2 \|a_{\Lambda_0}\|\, \|b_\Lambda\|_\Lambda.\] Since \((b_\Lambda)_\Lambda\) is a fixed local observable, \(\|b_\Lambda\|_\Lambda\) is uniformly bounded. Hence, \[\| [a_\Lambda, b_\Lambda] \|_\Lambda=O(1/N),\] and thus \[\lim_{\Lambda \nearrow \Gamma} \| [a_\Lambda, b_\Lambda] \|_\Lambda = 0.\] This shows that every \(\gamma\)-sequence \((a_\Lambda)_\Lambda\) belongs to \(\mathcal{C}^\infty\). Passing to equivalence classes via the canonical quotient map \[q : \prod_{\Lambda \Subset \Gamma} \mathcal{B}_\Lambda \;\to\; [\mathcal{B}]_\sim,\] we obtain \[[a_\Lambda]_\Lambda = q((a_\Lambda)_\Lambda) \in [\mathcal{C}]^\infty.\] ◻
Remark 3. The \(C^*\)-algebra \([\mathcal{B}]^\infty_\gamma\) properly contains the commutative \(C^*-\) subalgebra \([\mathcal{B}]_\pi^\infty\), that is, the algebra of equivalence classes generated by permutation symmetric sequences. In particular, \([\mathcal{B}]_\pi^\infty\cong C(S(\mathcal{B}))\) [9], [10], which is a direct consequence of the quantum De Finetti theorem. \(\blacksquare\)
Consider again the discrete set \(\Gamma \subset \mathbb{R}^d\). To each \(x \in \Gamma\) we associate a unital commutative \(C^*\)-algebra endowed with a Poisson bracket, \((\mathcal{A}_x, \{\cdot,\cdot\})\). This structure plays an important role in equilibrium thermodynamics within classical statistical mechanics, as it provides a natural framework for formulating the classical KMS condition [12].
For any finite subset \(\Lambda \subset \Gamma\), we denote the tensor product \(\bigotimes_{x \in \Lambda} \mathcal{A}_x\) by \(\mathcal{A}_\Lambda\), and equip it with the local supremum norm \(\|\cdot\|_\Lambda\). To define the Poisson bracket, we first fix a dense Poisson \(*\)-subalgebra \(\dot{\mathcal{A}_x} \subset \mathcal{A}_x\) and then define \(\dot{\mathcal{A}}_\Lambda\) accordingly for each finite subset \(\Lambda \Subset \Gamma\). Then, in a fashion similar to above, the algebra \[\dot{\mathcal{D}}^\infty:=\left\{ (a_\Lambda)_\Lambda \in \prod_{\Lambda\Subset\Gamma} \dot{\mathcal{A}}_\Lambda : \forall \quad (b_\Lambda)_\Lambda \in \dot{ \mathcal{A}}^\infty, \;\lim_{\Lambda \nearrow \Gamma} \|\{a_\Lambda, b_\Lambda\}_\Lambda\|_\Lambda = 0 \right\},\] where \(\dot{\mathcal{A}}^\infty\) denotes the Poisson algebra of all local sequences, is a commutative \(*\)-subalgebra of \(\prod_{\Lambda\Subset\Gamma} \mathcal{A}_\Lambda\). For a finite region \(\Lambda \Subset \Gamma\), we define \[\dot{\mathcal{D}}_\Lambda^\infty := \Big\{ (a_{\Lambda'})_{\Lambda'} \in \prod_{\Lambda' \Subset \Gamma} \dot{\mathcal{A}}_{\Lambda'} \;:\; \forall b_\Lambda \in \dot{\mathcal{A}}_\Lambda, \;\lim_{\Lambda' \nearrow \Gamma} \| \{ a_{\Lambda'}, b_\Lambda \}_{\Lambda'} \|_{\Lambda'} = 0 \Big\},\] where \(b_\Lambda\) is embedded canonically into \(\mathcal{A}_{\Lambda'}\) for \(\Lambda' \supseteq \Lambda\). Analogous to the algebra \(\mathcal{C}^\infty\), it follows that \[\dot{\mathcal{D}}^\infty = \bigcap_{\Lambda \Subset \Gamma} \dot{\mathcal{D}}_\Lambda^\infty.\] Moreover, we can complete \(\dot{\mathcal{D}}^\infty\) with respect to the norm \(\|\cdot\|:=\sup_{\Lambda\Subset\Gamma}\|\cdot\|_\Lambda\), yielding the \(C^*\)-algebra \[\mathcal{D}^\infty:=\overline{\dot{\mathcal{D}}^\infty}^{\|\cdot\|}.\] The pertinent quotient \[[\mathcal{D}]^\infty:= \mathcal{D}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda\] with \(\bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda\) the ideal of vanishing sequences, is the the classical analog of \([\mathcal{C}]^\infty\), and forms a \(C^*\)-subalgebra of the quotient \(C^*\)-algebra \([\mathcal{A}]_\sim.\) Note that this algebra can be also obtained from \(\dot{\mathcal{D}}^\infty\), as follows from \[\mathcal{D}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda=\overline{\dot{\mathcal{D}}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda}^{\|\cdot\|},\] with norm given by 5 .
Remark 4. We recall the tail-\(\sigma\)-algebra from classical statistical mechanics: \[\label{eq:tail95sigma} \mathscr{T}_\infty := \bigcap_{\Lambda \subset \mathbb{Z}^d} \mathscr{F}_{\Lambda^c},\tag{9}\] where \(\mathscr{F}_{\Lambda^c}\) is the \(\sigma\)-algebra of events that only depend on “spins” located outside \(\Lambda\). The algebra \([\mathcal{C}]^\infty\) resembles its quantum analog, canonically formalized in a \(C^*\)-algebraic way, i.e. without passing to any representation. \(\blacksquare\)
We have the following result, connecting \([\mathcal{D}]^\infty\) with tail-measurable functions.
Proposition 5. Let \(X\) be a connected finite dimensional symplectic manifold and consider the \(C^*\)-algebra \(\mathcal{A} = C_0(X)\). Let \(\Gamma\) be a countable index set and suppose \(\mathcal{A}_x = C_0(X)\) for all \(x \in \Gamma\). Define the set \[\Omega := \prod_{x \in \Gamma} X_x.\] Then, for any equivalence class \([a] \in [\mathcal{D}]^\infty\), there exists a unique bounded function \[f_a : \Omega \to \mathbb{C}\] which is measurable with respect to the tail \(\sigma\)-algebra \[\mathscr{T}_\infty := \bigcap_{\Lambda \Subset \Gamma} \sigma\big(\{\omega_x : x \in \Gamma \setminus \Lambda\}\big).\] Conversely, each function \(f : \Omega \to \mathbb{C}\) that is bounded and measurable with respect to the tail \(\sigma\)-algebra, induces a unique equivalence class \([a] \in [\mathcal{D}]^\infty\).
Proof. Note that \(\dot{\mathcal{D}}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda\) is dense in \([\mathcal{D}]^\infty\). Given an equivalence class \([a] \in [\mathcal{D}]^\infty\) and \(\epsilon>0\), find \([\dot{a}]\in \dot{\mathcal{D}}^\infty/ \bigoplus_{\Lambda\Subset\Gamma}\mathcal{A}_\Lambda\), such that \[\|[a]-[\dot{a}]\|=\limsup_{\Lambda'\nearrow\Gamma}\|a_{\Lambda'}-\dot{a}_{\Lambda'}\|_{\Lambda'}<\epsilon.\] Based on this, we first prove the result for \([\dot{a}]\). To this end, let us fix a representative net \((\dot{a}_{\Lambda'})_{\Lambda'}\in\dot{\mathcal{D}}^\infty\). We can view it as a function on \[\Omega = \prod_{x \in \Gamma} X_x\] via the canonical projection \(\omega \mapsto \omega|_{\Lambda'}\), i.e., \[\dot{a}_{\Lambda'}(\omega) := \dot{a}_{\Lambda'}(\omega|_{\Lambda'}) \quad \text{for all } \omega \in \Omega.\] We then define a function \(f_{\dot{a}} : \Omega \to \mathbb{C}\) by \[f_{\dot{a}}(\omega) := \limsup_{\Lambda'\nearrow\Gamma}\dot{a}_{\Lambda'}(\omega).\] This function is well defined and hence unique for the given equivalence class: if \((\dot{b}_{\Lambda'})_{\Lambda'}\) is another representative, then \[\lim_{\Lambda'\nearrow\Gamma}\|\dot{b}_{\Lambda'} - \dot{a}_{\Lambda'}\|_{\Lambda'}=0,\] and hence, for all \(\omega\), \[|f_{\dot{a}}(\omega)-f_{\dot{b}}(\omega)|\leq \limsup_{\Lambda'\nearrow\Gamma}\|\dot{a}_{\Lambda'} - \dot{b}_{\Lambda'}\|_{\Lambda'}=\lim_{\Lambda'\nearrow\Gamma}\|\dot{a}_{\Lambda'} - \dot{b}_{\Lambda'}\|_{\Lambda'}=0.\] The function \(f_{\dot{a}}\) is bounded: \[\|f_{\dot{a}}\|_\infty := \sup_{\omega \in \Omega} |f_{\dot{a}}(\omega)|<\infty\] because of the uniform boundedness of the sequence \((\|\dot{a}_\Lambda\|)_\Lambda\in \ell^\infty(\Gamma)\). Thus, \(f_{\dot{a}}\in \ell^\infty(\Omega)\).
We now prove that \(f_{\dot{a}}\) is tail-measurable. Fix any \(\Lambda_0\Subset\Gamma\). Note first that if \(g\) is any smooth compactly supported local
self-adjoint element with \(\text{supp}(g)\subset\Lambda_0\), its Hamiltonian vector field \(X_{g}\) is complete, and the (globally defined for all \(t\))
flow \(\varphi_t^g\) only acts on the local coordinates \(\omega_{\Lambda_0}\in\Omega_\Lambda\). Let \(g_{\Lambda'}\) be the canonical embedding of \(g\) into \(\dot{\mathcal{A}}_{\Lambda'}\). Likewise, each \(\dot{a}_{\Lambda'}\) with \(\Lambda'\supseteq\Lambda_0\)
is a smooth local observable depending only on \(\omega_{\Lambda'}\). The chain rule then gives \[\frac{d}{dt}\big(\dot{a}_{\Lambda'}(\varphi_t^{g_{\Lambda'}}(\omega_{\Lambda'}))\big) = \{\dot{a}_{\Lambda'},g_{\Lambda'}\}(\varphi_t^{g_{\Lambda'}}(\omega_{\Lambda'})).\] Integrating in \(t\) we obtain \[\dot{a}_{\Lambda'}(\varphi_t^{g_{\Lambda'}}(\omega_{\Lambda'})) - \dot{a}_{\Lambda'}(\omega_{\Lambda'}) = \int_0^t
\{\dot{a}_{\Lambda'},g_{\Lambda'}\}(\varphi_s^{g_{\Lambda'}}(\omega_{\Lambda'}))\,ds.\] This implies the uniform estimate \[\sup_{\omega_{\Lambda'}}\bigg{|}\dot{a}_{\Lambda'}(\varphi_t^{g_{\Lambda'}}(\omega_{\Lambda'})) - \dot{a}_{\Lambda'}(\omega_{\Lambda'})\bigg{|} \le |t|\,
\|\{\dot{a}_{\Lambda'},g_{\Lambda'}\}\|_{\Lambda'},\] where for \(\omega \in \Omega\), we have written \(\omega_{\Lambda'} := \omega|_{\Lambda'}\). Taking \(\limsup_{\Lambda'\nearrow\Gamma}\) and using \((\dot{a}_{\Lambda'})_{\Lambda'} \in \dot{\mathcal{D}}^\infty\), the right-hand side tends to \(0\).
This implies that for any \(t\) in a compact set and any \(\omega_{\Lambda'}\in\Omega_{\Lambda'}\) \[\limsup_{\Lambda'\nearrow\Gamma}
\bigg{|}\dot{a}_{\Lambda'}(\varphi_t^{g_{\Lambda'}}(\omega_{\Lambda'})) - \dot{a}_{\Lambda'}(\omega_{\Lambda'})\bigg{|} = 0.\] By the definition of \(f_{\dot{a}}\) this entails \[f_{\dot{a}}(\Phi_t^{g}(\omega)) = f_{\dot{a}}(\omega) ,\quad \text{for all t\in\mathbb{R}},\] where \(\Phi_t^{g}(\omega):=(\varphi_t^g(\omega_{\Lambda_0}),\omega_{\Lambda_0^c})\). This
expression is well defined, since \(g\) is localized on \(\Omega_{\Lambda_0}\), and hence its flow acts only on coordinates inside \(\Lambda_0\).
Equivalently, for any configuration \(\omega\in\Omega\), \[f_{\dot{a}}(\varphi_t^{g}(\omega_{\Lambda_0}),\omega_{\Lambda_0^c}) = f_{\dot{a}}(\omega_{\Lambda_0},\omega_{\Lambda_0^c}) ,\quad
\text{for all t\in\mathbb{R}},\] This \(f_{\dot{a}}\) is invariant under local Hamiltonian flows supported in \(\Lambda_0\). To conclude, we take \(\omega,
\rho\in \Omega\) with \(\omega|_{\Lambda_0^c} = \rho|_{\Lambda_0^c}\). By standard constructions, the group generated by compactly supported Hamiltonian diffeomorphisms acts transitively on a connected, symplectic
manifold. Hence, any two local configurations can be connected by a finite composition of local Hamiltonian flows supported in \(\Lambda_0\). Hence, there exists a finite sequence of smooth local Hamiltonians \(g_1,\dots,g_n\) supported in \(\Lambda_0\) and times \(t_1,\dots,t_n\) such that \[\rho_{\Lambda_0} = \varphi_{t_n}^{g_n} \circ \cdots
\circ \varphi_{t_1}^{g_1}(\omega_{\Lambda_0}).\] Since \(f_{\dot{a}}\) is invariant under each \(\varphi_{t_j}^{g_j}\), we obtain \[f_{\dot{a}}(\rho_{\Lambda_0},\rho_{\Lambda_0^c}) = f_{\dot{a}}(\omega_{\Lambda_0},\omega_{\Lambda_0^c}),\] and hence \(f_{\dot{a}}\) does not depend on coordinates inside any local region.
Thus, \[\forall \Lambda_0 \Subset \Gamma,\;\forall \omega, \rho \in \Omega,\;
\omega|_{\Lambda_0^c} = \rho|_{\Lambda_0^c} \implies f_{\dot{a}}(\omega) = f_{\dot{a}}(\rho).\] This is exactly the definition of tail-measurability. Finally, note that since \[\|f_a-f_{\dot{a}}\|_\infty\leq
\limsup_{\Lambda'\nearrow\Gamma}\|a_{\Lambda'} -\dot{a}_{\Lambda'}\|_{\Lambda'}<\epsilon,\] the same result holds for \(f_a\), i.e. the function induced by the equivalence class \([a]\). This proves the forward implication.
For the converse, let \(f\) be \(\mathscr{T}_\infty\)-measurable. By definition, \(f\) is measurable with respect to every \(\mathscr{F}_{\Lambda^c}\). This means that for every finite subset \(\Lambda \Subset \Gamma\) there exists a \(\mathscr{F}_{\Lambda^c}\)-measurable function
\[f_\Lambda : \prod_{x \in \Gamma \setminus \Lambda} X_x \to \mathbb{C}\] such that for all \(\omega = (\omega_x)_{x \in \Gamma} \in \Omega\), \[f(\omega) =
f_\Lambda\big( (\omega_x)_{x \in \Gamma \setminus \Lambda} \big).\] Define the sequence \[a = (a_{\Lambda})_{\Lambda \Subset \Gamma}, \quad \text{with} \quad a_{\Lambda} := f_{\Lambda}.\] Because \(f\) is bounded and depends only on coordinates outside \(\Lambda\), each \(a_{\Lambda}\) is bounded and \(a_{\Lambda} \in
\mathcal{A}_{\Lambda^c}\), where the complement is taken within the full \(C^*-\)product. By construction, \(a = (a_\Lambda)_{\Lambda}\) lies in \(\mathcal{D}^\infty\), since each \(a_\Lambda \in \mathcal{A}_{\Lambda^c}\), i.e. it does not depends on \(\Lambda\). Hence, it eventually Poisson-commutes with
all local sequences. More precisely, for any finite \(\Lambda' \Subset \Gamma\) and any \(b \in \mathcal{A}_{\Lambda'}\), there exists \(\Lambda_0 \supseteq
\Lambda'\) such that for all \(\Lambda \supseteq \Lambda_0\), \[\{ a_\Lambda, b \}_\Lambda = 0,\] implying that \[\lim_{\Lambda \nearrow \Gamma} \| \{
a_\Lambda, b \}_\Lambda \|_\Lambda = 0,\] proving that the sequence \(a = (a_\Lambda)_{\Lambda}\) indeed lies in \(\mathcal{D}^\infty\), and therefore defines an equivalence class
\([a] \in [\mathcal{D}]^\infty\). To prove it is unique, we proceed by contradiction. Assume \([a] \neq [a']\) in \([\mathcal{D}]^\infty\). Take two
representatives \[a = (a_\Lambda)_{\Lambda \Subset \Gamma}
\quad \text{and} \quad
a' = (a'_\Lambda)_{\Lambda \Subset \Gamma}\] associated with the same \(\mathscr{T}_\infty\)-measurable function \(f\) through the construction above. Note that, by
definition, it follows that \[\limsup_{\Lambda \nearrow \Gamma} \| a_\Lambda - a'_\Lambda \|_\Lambda > 0.\] Hence, there exists \(\varepsilon > 0\) and an increasing sequence of
finite subsets \(\Lambda_n \Subset \Gamma\) such that \[\| a_{\Lambda_n} - a'_{\Lambda_n} \|_{\Lambda_n} \ge \varepsilon \quad \text{for all } n.\] But by construction each \(a_\Lambda\) and \(a'_\Lambda\) is the \(\mathscr{F}_{\Lambda^c}\)-measurable representative of the same function \(f\) on
the coordinates \(\Lambda^c\). Fix any finite subset \(\Lambda_0 \subset \Gamma\). Since \(f\) is \(\mathscr{T}_\infty\)-measurable, there exists \(N\) such that for all \(n \ge N\), if \(\Lambda_n \supset \Lambda_0\), then
\[a_{\Lambda_n}(\omega) = a'_{\Lambda_n}(\omega) \quad \text{for all } \omega \in \Omega.\] In particular, the sup norm for these approximations satisfies \[\| a_{\Lambda_n} -
a'_{\Lambda_n} \|_{\Lambda_n} = 0\] for all sufficiently large \(n\), which contradicts the assumption that \[\| a_{\Lambda_n} - a'_{\Lambda_n} \|_{\Lambda_n} \ge
\varepsilon.\] Hence, \([a]\) and \([a']\) define the same equivalence class in \([\mathcal{D}]^\infty\). ◻
This work opens new avenues for future research in this field. Some potential directions are summarized below.
Investigate the state space of the non-commutative \(C^*\)-algebra \([\mathcal{C}]^\infty\), in particular whether it forms a Choquet simplex (Bauer, Poulsen, etc.) and characterize extremal states via a suitable notion of ergodicity [13].
Study the dynamics of the algebra, including automorphisms and strong continuity, with applications to quantum spin systems where each local \(C^*\)-algebra is a matrix algebra.
Explore the Kubo-Martin-Schwinger (KMS) condition for \([\mathcal{B}]_\gamma^\infty\), the existence and properties of KMS states, and their relation to translation-invariant states on the quasi-local algebra [1]–[3].
For the commutative \(C^*\)-algebra \([\mathcal{D}]^\infty\), examine the classical KMS condition via the Poisson bracket and compare it with the Dobrushin-Lanford-Ruelle (DLR) equilibrium condition [12], [14].
The author is grateful to Aernout van Enter, Jean-Bernard Bru and Nicolo’ Drago for many helpful comments and suggestions.
This notion of convergence should not be confused with convergence in the sense of van Hove.↩︎