Deduction and definability in infinite statistical systems

Abstract

Classical accounts of intertheoretic reduction involve two pieces: first, the new terms of the higher-level theory must be definable from the terms of the lower-level theory, and second, the claims of the higher-level theory must be deducible from the lower-level theory along with these definitions. The status of each of these pieces becomes controversial when the alleged reduction involves an infinite limit, as in statistical mechanics. Can one define features of or deduce the behavior of an infinite idealized system from a theory describing only finite systems? In this paper, I change the subject in order to consider the motivations behind the definability and deducibility requirements. The classical accounts of intertheoretic reduction are appealing because when the definability and deducibility requirements are satisfied there is a sense in which the reduced theory is forced upon us by the reducing theory and the reduced theory contains no more information or structure than the reducing theory. I will show that, likewise, there is a precise sense in which in statistical mechanics the properties of infinite limiting systems are forced upon us by the properties of finite systems, and the properties of infinite systems contain no information beyond the properties of finite systems.

Appendix: Proofs of Results

This appendix contains proofs of all results in Sects. 4 and 5.

Proposition 1

Given any Banach space X, its dual $X^*$ is complete in the weak* topology as a locally convex vector space.

Proof

Suppose $\{y_\beta \}$ is a Cauchy net in $X^*$. Define $y:X\rightarrow \mathbb {C}$ by

$$\begin{aligned} y(x) = \lim y_\beta (x) \end{aligned}$$

for all $x\in X$. We know that this limit exists because for any $x\in X$, $y_\beta (x)$ is a Cauchy net in $\mathbb {C}$, which means it must converge because $\mathbb {C}$ is complete.

Now we must show that $y\in X^*$, i.e., that y is linear and bounded. The functional y is linear because for any $x,x'\in X$ and $\alpha \in \mathbb {C}$,

$$\begin{aligned} y(x + \alpha x') = \lim y_\beta (x + \alpha x') = \lim y_\beta (x) + \alpha \lim y_\beta (x') = y(x) + \alpha y(x') \end{aligned}$$

Notice that because $|y_\beta (x)|$ is bounded for each $x\in X$, it follows from the principle of uniform boundedness that $||y_\beta ||$ is bounded (Reed and Simon 1980). Hence, y is bounded with norm $||y||\le \sup _{\beta }||y_\beta ||$. Finally, we must show that $y_\beta$ converges to y in the weak* topology on $X^*$. But this holds by construction because for any $x\in X$, $(y-y_\beta )(x)$ converges to zero in $\mathbb {C}$. $\square$

Proposition 2

Suppose X is a Banach space and $J_X:X\rightarrow X^{**}$ is the canonical evaluation embedding of X in its bidual. Suppose we are given another faithful linear embedding $K:X\rightarrow Y$ of X in a complete locally convex vector space Y such that K(X) is dense in Y in the locally convex vector space topology on Y. Suppose, in addition, that K is a homeomorphism from X to K(X) in the weak topology on X and the subspace topology on K(X) generated by the locally convex vector space topology on Y. Then there is a vector space isomorphism $\varphi :Y\rightarrow X^{**}$ that is a homeomorphism in the locally convex vector space topology on Y and the weak* topology on $X^{**}$ and such that $J_X = \varphi \circ K$.

Proof

Suppose $K:X\rightarrow Y$ is such an embedding. By Corollary 1.2.3 of Kadison and Ringrose (1997, p. 15), the maps $\varphi _0 = J_X\circ K^{-1}$ and $\psi _0 = K\circ J_X^{-1}$ extend uniquely to continuous linear maps $\varphi : Y\rightarrow X^{**}$ and $\psi : X^{**}\rightarrow Y$ in the weak* topology on $X^{**}$ and the locally convex vector space topology on Y. Since $\varphi _0\circ \psi _0$ and $\psi _0\circ \varphi _0$ are the identity operators on $J_X(X)$ and K(X), respectively, it follows that $\varphi \circ \psi$ and $\psi \circ \varphi$ are the identity operators on $X^{**}$ and Y, respectively. Thus, $\varphi$ is an isomorphism and a homeomorphism in the weak* topology on $X^{**}$ and the locally convex vector space topology on Y. By construction, we have $\varphi \circ K = \varphi _0\circ K = J_X\circ K^{-1}\circ K = J_X$.

$\square$

Proposition 3

Suppose $\alpha :\mathfrak {A}\rightarrow \mathfrak {B}$ is a *-homomorphism between C*-algebras $\mathfrak {A}$ and $\mathfrak {B}$. Then there is a unique weak* continuous extension $\tilde{\alpha }:\mathfrak {A}^{**}\rightarrow \mathfrak {B}^{**}$ such that $\tilde{\alpha }\circ J_\mathfrak {A} = J_\mathfrak {B}\circ \alpha$, where $J_\mathfrak {A}:\mathfrak {A}\rightarrow \mathfrak {A}^{**}$ and $J_\mathfrak {B}:\mathfrak {B}\rightarrow \mathfrak {B}^{**}$ are the canonical evaluation maps.

Proof

First, we show that $\alpha$ is continuous in the weak Banach space topologies on $\mathfrak {A}$ and $\mathfrak {B}$. Suppose we have a net $\{A_\beta \}\subseteq \mathfrak {A}$ such that $A_\beta \rightarrow A$ weakly. Then for all $\rho \in \mathfrak {A}^*$, $\rho (A_\beta )\rightarrow \rho (A)$. Consider the net $\{\alpha (A_\beta )\}\subseteq \mathfrak {B}$. For any $\sigma \in \mathfrak {B}^*$, $\sigma \circ \alpha \in \mathfrak {A}^*$, so it follows that $\sigma \circ \alpha (A_\beta )\rightarrow \sigma \circ \alpha (A)$. Hence, $\alpha (A_\beta )\rightarrow \alpha (A)$ weakly, and it follows that $\alpha$ is weakly continuous.

It follows from Corollary 1.2.3 of Kadison and Ringrose (1997, p. 15) that the map $J_\mathfrak {B}\circ \alpha \circ J_\mathfrak {A}^{-1}: J_\mathfrak {A}(\mathfrak {A})\rightarrow \mathfrak {B}^{**}$ extends uniquely to a weak* continuous map $\tilde{\alpha }: \mathfrak {A}^{**}\rightarrow \mathfrak {B}^{**}$. By construction,

$$\begin{aligned} \tilde{\alpha }\circ J_\mathfrak {A} = J_\mathfrak {B}\circ \alpha \circ J_\mathfrak {A}^{-1}\circ J_\mathfrak {A} = J_\mathfrak {B}\circ \alpha \end{aligned}$$

$\square$

Proposition 4

G is not full, i.e., G forgets structure.

Proof

By Proposition 5 below, F and G form an adjunction, and we know G is faithful. So it follows from Proposition 3.4.1 of Borceux (1994, p. 114) that G is full only if the counit of the adjunction $\tilde{1}: G\circ F\rightarrow 1_{\mathbf{W*-Alg }}$ is a natural isomorphism.

The counit of the adjunction has as its component on any object $\mathfrak {R}$ in W*-Alg the unique continuous extension $\tilde{1}_{\mathfrak {R}}:\mathfrak {R}^{**}\rightarrow \mathfrak {R}$ of the identity arrow $1_{\mathfrak {R}}: \mathfrak {R}\rightarrow \mathfrak {R}$ in the weak topology on the domain and the weak* topology on the codomain, which exists by Corollary 1.2.3 of Kadison and Ringrose (1997, p. 15) since $\mathfrak {R}$ is weak* complete. Notice that $\tilde{1}_\mathfrak {R}$ is not the identity map from $\mathfrak {R}^{**}$ to itself, because we require continuity in the weak* topology on the codomain rather than the weak topology, so that the codomain is already complete and even after continuously extending $1_\mathfrak {R}$ in the relevant topologies, we get a map into the codomain $\mathfrak {R}$. This map $\tilde{1}_\mathfrak {R}$ satisfies the universal property for counits, i.e., given any $\mathfrak {A}$ in C*-Alg, any $\mathfrak {R}$ in W*-Alg and arrow $\tilde{\alpha }:G(\mathfrak {A})\rightarrow \mathfrak {R}$, there is a unique arrow $\alpha : \mathfrak {A}\rightarrow F(\mathfrak {R})$ such that $\tilde{\alpha } = \tilde{1}_{\mathfrak {R}}\circ G(\alpha )$. Here, $\alpha$ is just the restriction of $\tilde{\alpha }$ from $G(\mathfrak {A}) = \mathfrak {A}^{**}$ to $\mathfrak {A}$, i.e., $\alpha = \tilde{\alpha }_{|\mathfrak {A}}$.

Indeed, $\tilde{1}$ is a natural transformation because for any two objects $\mathfrak {R}_1$ and $\mathfrak {R}_2$ in W*-Alg and any arrow $\alpha : \mathfrak {R}_1\rightarrow \mathfrak {R}_2$, we know that $\alpha \circ \tilde{1}_{\mathfrak {R}_1} = \tilde{1}_{\mathfrak {R}_2}\circ G\circ F(\alpha )$ because there is a unique weakly continuous extension of $\alpha$ from $\mathfrak {R}_1$ to $G\circ F(\mathfrak {R}_1) = \mathfrak {R}_1^{**}$ by Corollary 1.2.3 of Kadison and Ringrose (1997, p. 15).

In general, $\tilde{1}_{\mathfrak {R}}$ will not be an isomorphism. For, if $\mathfrak {R}\ncong \mathfrak {R}^{**}$ (as is the case for infinite dimensional W*-algebras), since $1_{\mathfrak {R}}:\mathfrak {R}\rightarrow \mathfrak {R}$ is surjective and $J_{\mathfrak {R}}(\mathfrak {R})\subsetneq \mathfrak {R}^{**}$, it follows that $\tilde{1}_{\mathfrak {R}}:\mathfrak {R}^{**}\rightarrow \mathfrak {R}$ cannot be one-to-one and so cannot be an isomorphism. Thus, the counit $\tilde{1}$ is not a natural isomorphism, and so G is not full. $\square$

Proposition 5

F and G form an adjunction, with left adjoint G, right adjoint F and unit J.

Proof

First, we know that J is a natural transformation from $F\circ G$ to $1_{\mathbf{C*-Alg }}$ because $F\circ G(\mathfrak {A})$ is just $\mathfrak {A}^{**}$ considered as an object in C*-Alg. So for any two objects $\mathfrak {A}$ and $\mathfrak {B}$ in C*-Alg and any arrow $\alpha :\mathfrak {A}\rightarrow \mathfrak {B}$, it follows by Proposition 3 that

$$\begin{aligned} J_\mathfrak {B}\circ 1_{{\mathbf {C^{*}-Alg}}}(\alpha ) = F\circ G(\alpha )\circ J_{\mathfrak {A}} \end{aligned}$$

J serves as the unit because for any objects $\mathfrak {A}$ in C*-Alg and $\mathfrak {R}$ in W*-Alg and arrow $\alpha : \mathfrak {A}\rightarrow F(\mathfrak {R})$, there is a unique arrow $\tilde{\alpha }: \mathfrak {A}^{**}\rightarrow \mathfrak {R}$ such that $F(\tilde{\alpha })\circ J_{\mathfrak {A}} = \alpha$. The arrow $\tilde{\alpha }$ is given by the unique weakly continuous extension of $\alpha$ to the weak completion $G(\mathfrak {A}) = \mathfrak {A}^{**}$.

$\square$

Proposition 6

Let $\mathfrak {R}$ be a W*-algebra and let

$$\begin{aligned} I = \{A\in \mathfrak {R}: \rho (A) = 0\text { for all }\rho \in \mathfrak {R}_*\} \end{aligned}$$

There is a C*-algebra $\mathfrak {A}$ such that $\mathfrak {R}\cong \mathfrak {A}^{**}$ iff

$$\begin{aligned} \mathfrak {R}_* \cong \left\{ \omega \in \mathfrak {R}^*:\text { if } A\in I, \text { then } \omega (A) = 0\right\} \end{aligned}$$

(2)

Proof

($\Leftarrow$) Suppose that $\mathfrak {R}$ satisfies Eq. 1. By Proposition 2.11.8 of Dixmier (1977, p. 63) and Corollary 1.8.3 of Dixmier (1977, p. 21), it follows that $\mathfrak {A} := \mathfrak {R}/I$ is a C*-algebra such that $\mathfrak {A}^*\cong \mathfrak {R}_*$. Hence, we know that $\mathfrak {A}^{**}$ is isomorphic to $\mathfrak {R}$ as a Banach space. It suffices to show that the canonical embedding $J_\mathfrak {A}:\mathfrak {A}\rightarrow \mathfrak {A}^{**}$ induces a map (which we will also call $J_\mathfrak {A}$ since no ambiguity will result) from $\mathfrak {R}/I$ to $\mathfrak {R}$ that preserves multiplication in order to show that $\mathfrak {A}^{**}$ and $\mathfrak {R}$ are *-isomorphic. We can choose the isomorphisms so that $J_\mathfrak {A}$ induces the map

$$\begin{aligned} (A + I)\in \mathfrak {R}/I\mapsto A\in \mathfrak {R} \end{aligned}$$

We lose no generality in choosing a representative $A\in A+ I$ in this way (although we require the axiom of choice). It is easy to check that this map is a faithful *-homomorphism whose range is weak* dense in $\mathfrak {R}$. It follows that $J_\mathfrak {A}$ extends to a *-isomorphism from $\mathfrak {A}^{**}$ to $\mathfrak {R}$.

($\Rightarrow$) Suppose that $\mathfrak {R}\cong \mathfrak {A}^{**}$ for some C*-algebra $\mathfrak {A}$. We know from Corollary 1.13.3 of Sakai (1971, p. 30) that $\mathfrak {R}_*\cong \mathfrak {A}^*$. Let

$$\begin{aligned} I_\mathfrak {A} = \left\{ A\in \mathfrak {A}^{**}: \rho (A) = 0\text { for all } \rho \in \mathfrak {A}^*\right\} \end{aligned}$$

Then $\mathfrak {R}_*\cong \mathfrak {A}^*\cong (\mathfrak {A}^{**}/I_{\mathfrak {A}})^*$, and by Proposition 2.11.8 of Dixmier (1977, p. 63), we know that

$$\begin{aligned} (\mathfrak {A}^{**}/I_\mathfrak {A})^* \cong \left\{ \omega \in \mathfrak {A}^{***}:\text { if } A\in I_\mathfrak {A}, \text { then } \omega (A) = 0.\right\} \end{aligned}$$

which implies that $\mathfrak {R}_*$ satisfies Eq. 1. $\square$

Proposition 7

Let $\mathfrak {R}$ be a W*-algebra. If $\mathfrak {A}$ and $\mathfrak {B}$ are two C*-algebras such that $\mathfrak {A}^{**}\cong \mathfrak {R}\cong \mathfrak {B}^{**}$, then $\mathfrak {A}$ is *-isomorphic to $\mathfrak {B}$.

Proof

We know from Corollary 1.13.3 of Sakai (1971, p. 30) that $\mathfrak {A}^*\cong \mathfrak {B}^*$. As before, let

$$\begin{aligned} I_\mathfrak {A}= & {} \left\{ A\in \mathfrak {A}^{**}: \rho (A) = 0\text { for all } \rho \in \mathfrak {A}^*\right\} \\ I_\mathfrak {B}= & {} \left\{ B\in \mathfrak {B}^{**}: \rho (B) = 0\text { for all } \rho \in \mathfrak {B}^*\right\} \end{aligned}$$

Again, we know from Proposition 2.11.8 of Dixmier (1977, p. 63) that $(\mathfrak {A}^{**}/I)^*\cong \mathfrak {A}^*$ and $(\mathfrak {B}^{**}/I)^*\cong \mathfrak {B}^*$. Hence, the canonical surjective *-homomorphisms $\mathfrak {A}^{**}\rightarrow \mathfrak {A}^{**}/I$ and $\mathfrak {B}^{**}\rightarrow \mathfrak {B}^{**}/I$ can serve to define surjective *-homomorphisms from $\mathfrak {R}$ to the C*-algebras $\mathfrak {A}$ and $\mathfrak {B}$, who have isomorphic dual spaces. It follows from Theorem 2 of Feintzeig (2017a) that $\mathfrak {A}\cong \mathfrak {B}$. $\square$

Proposition 8

There are W*-algebras that are not whole.

Proof

Let $\mathfrak {R}:=L^\infty (\mathbb {R})$ be the algebra of equivalence classes of bounded Borel measurable complex-valued functions on the real line that differ only on sets of Lebesgue measure zero, where the algebraic structure is defined by pointwise operations. We know $\mathfrak {R}$ has no pure normal states (see, e.g. Halvorson 2001) (and any W*-algebra without pure normal states will suffice for the rest of the proof).²⁰ Suppose, for contradiction, that $\mathfrak {R}$ is whole. Then, by Proposition 6, we know $\mathfrak {R}_* = \mathfrak {A}^*$ is the dual space to some C*-algebra $\mathfrak {A}$. By the Banach-Alaoglu theorem (Corollary 1. 6.6 of Kadison and Ringrose 1997, p. 46) and the Krein-Milman theorem (Theorem 1.4.3 of Kadison and Ringrose 1997, p. 32), the state space of $\mathfrak {A}$, which is the normal state space of $\mathfrak {R}$, must contain pure states, which yields a contradiction. $\square$

Proposition 9

$\overline{G}$ is a categorical equivalence between C*-Alg and wW*-Alg.

Proof

$\overline{G}$ is faithful because G is faithful. Proposition 6 shows that $\overline{G}$ is essentially surjective. To show that $\overline{G}$ is full, consider any arrow $\alpha : \mathfrak {R}_1\rightarrow \mathfrak {R}_2$ in wW*-Alg, where $\mathfrak {A}_1$ and $\mathfrak {A}_2$ are the unique C*-algebras such that $\mathfrak {A}_1^{**}\cong \mathfrak {R}_1$ and $\mathfrak {A}_2^{**}\cong \mathfrak {R}_2$. To simplify notation, let $J_1:=J_{\mathfrak {A}_1}$ and $J_2:=J_{\mathfrak {A}_2}$. Since $\alpha$ is a *-whomomorphism, we know that the restriction $\alpha _{|J_{1}(\mathfrak {A}_1)}$ of $\alpha$ to $J_{1}(\mathfrak {A}_1)$ is a *-homomorphism from the C*-algebra $J_{1}(\mathfrak {A}_1)$ to the C*-algebra $J_{2}(\mathfrak {A}_2)$. It follows that $\alpha = \tilde{\alpha }_{|J_1(\mathfrak {A}_1)} = \overline{G}(\alpha _{|J_1(\mathfrak {A}_1)})$, where $\tilde{\alpha }_{|J_1(\mathfrak {A}_1)}: \mathfrak {R}_1\rightarrow \mathfrak {R}_2$ is the unique weak* continuous extension of $\alpha _{|J_1(\mathfrak {A}_1)}$ by Proposition 3. $\square$