Quasilocal C*-Algebras and Inductive Limits

The quasilocal C*-algebra is the norm completion of all bounded-region observables, assembled with their inclusion maps rather than with a preferred Hilbert space. Its universal property makes states and symmetries representation-independent. The construction works only when the local C*-norms and bonding maps are compatible; it is not a tensor product of spatial subsystems.

Required background. Haag–Kastler Nets and Locality supplies the directed regional assignment. Banach and Hilbert Spaces, Completion, and Riesz Representation supplies norm completion, and Equivalence, Uniqueness, and Comparison Notions supplies the universal-property notion of uniqueness used below.

Helpful background. Operator Algebras and Positive Functionals reviews C*-homomorphisms and states.

The C*-inductive limit

Let $(I,\preceq)$ be directed and let $\mathcal A_i$ be unital C*-algebras. For $i\preceq j$, suppose

$$\iota_{ji}:\mathcal A_i\longrightarrow\mathcal A_j$$

is an injective unital star homomorphism, with $\iota_{ii}=\mathrm{id}$ and $\iota_{ki}=\iota_{kj}\circ\iota_{ji}$. Injective C*-homomorphisms are isometric. Consequently, if $A_i$ and $A_j$ represent the same eventual observable, they have the same norm.

The algebraic direct limit consists of equivalence classes $[i,A_i]$, where

$$(i,A_i)\sim(j,A_j)$$

when there is $k\succeq i,j$ such that $\iota_{ki}(A_i)=\iota_{kj}(A_j)$. Addition and multiplication are performed after moving two representatives to a common upper stage. Coherence makes those operations independent of that stage, and

$$\lVert[i,A_i]\rVert=\lVert A_i\rVert$$

is well defined. Completing gives $\mathcal A_{\mathrm{ql}}=\varinjlim_i\mathcal A_i$.

Equivalently, the limit is characterized by canonical maps $\iota_i:\mathcal A_i\to\mathcal A_{\mathrm{ql}}$: every compatible family of star homomorphisms $\phi_i:\mathcal A_i\to\mathcal B$ factors uniquely through a star homomorphism $\phi:\mathcal A_{\mathrm{ql}}\to\mathcal B$. This is uniqueness up to a unique isomorphism respecting all local embeddings, not equality of presentations.

For a net over bounded regions, directedness follows because two bounded regions lie in a larger bounded region. Then

$$\mathcal A_{\mathrm{loc}}=\bigcup_{O\Subset M}\mathcal A(O), \qquad \mathcal A_{\mathrm{ql}}=\overline{\mathcal A_{\mathrm{loc}}}^{\lVert\cdot\rVert}.$$

This standard AQFT completion is stated explicitly in Fewster and Rejzner 2020, § 4.1, pp. 13–15 and originates in the net framework of Haag and Kastler 1964, §§ 2–3, pp. 850–856.

States and automorphisms on the limit

A compatible family of local states satisfies

$$\omega_j\circ\iota_{ji}=\omega_i.$$

It defines $\omega([i,A_i])=\omega_i(A_i)$ on the algebraic limit. Positivity and $\lvert\omega(A)\rvert\leq\lVert A\rVert$ make the extension to $\mathcal A_{\mathrm{ql}}$ unique. Conversely, every quasilocal state restricts to a compatible family. Local density matrices are neither required nor generally available.

Suppose a spacetime transformation $g$ carries regions to regions and gives isomorphisms $\alpha_{g,O}:\mathcal A(O)\to\mathcal A(gO)$ that commute with inclusions. The universal property supplies a unique automorphism $\alpha_g$ of $\mathcal A_{\mathrm{ql}}$. Strong or norm continuity of $g\mapsto\alpha_g$ is an additional analytic condition; it is not a consequence of the algebraic factorization alone.

First application: completing the free Weyl net

Canonical Quantization: Algebra, Representation, and State supplies the Weyl relations and the separation between the abstract algebra and its Fock representations.

For the massive scalar, let $\mathcal A(O)$ be generated by $W([f])$ with $\operatorname{supp}f\subset O$. The global symplectic space is the union of the local symplectic subspaces, so every Weyl generator occurs in some bounded double cone. The compatible local algebras therefore embed in the global Weyl C*-algebra, and

$$\mathcal A_{\mathrm{ql}} = \overline{\bigcup_{O}\mathcal A(O)}^{\lVert\cdot\rVert}$$

is its quasilocal realization. No vacuum representation has entered.

For a translation $a$, set $f_a(x)=f(x-a)$. Since $E(f_a,g_a)=E(f,g)$,

$$\alpha_a(W([f]))=W([f_a])$$

preserves the Weyl relations and the C*-norm, maps $\mathcal A(O)$ onto $\mathcal A(O+a)$, and extends uniquely to an automorphism of the completion. Vacuum and thermal states can now be placed on the same abstract algebra even when their GNS representations are inequivalent.

Failure test: incompatible embeddings

An injective C*-homomorphism cannot be nonisometric. Thus a proposed bonding map that changes the norm of an observable has already failed to be an injective C*-homomorphism. A concrete loss of locality occurs with the coordinate projection

$$q:\mathbb C\oplus\mathbb C\longrightarrow\mathbb C, \qquad q(z,w)=z.$$

It is a unital star homomorphism but not injective: the observable $(0,1)$ disappears. A direct limit may still be formed in a broader category, but it no longer contains the first local algebra faithfully. Likewise, if two paths from $\mathcal A_i$ to $\mathcal A_k$ disagree, the equivalence relation depends on the chosen path and no claimed canonical local inclusion results.

The strongest survivor is an algebraic quotient or a presentation-dependent completion. Calling it the quasilocal algebra of the original net requires faithful coherent embeddings and compatible C*-norms.

Independent check

For every diamond $i\preceq j_1,j_2\preceq k$, compare

$$\iota_{kj_1}\iota_{j_1i}(A) \quad\text{and}\quad \iota_{kj_2}\iota_{j_2i}(A)$$

on a generating set. Then verify norm equality along each inclusion. For the Weyl system, it suffices to check that both paths send $W([f])$ to the same generator and preserve its Weyl products. This exact finite check detects both incoherent embeddings and accidental quotienting.

Exercises

Show that a compatible family of local states defines a bounded functional of norm one on the algebraic direct limit.

Solution

Compatibility makes $\omega([i,A_i])=\omega_i(A_i)$ independent of the representative. Each local state obeys $\lvert\omega_i(A_i)\rvert\leq\lVert A_i\rVert$, so $\lvert\omega(A)\rvert\leq\lVert A\rVert$ on the limit. Since $\omega(1)=1$, its norm is exactly one. The bound gives the unique continuous extension to the C*-completion.

References

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI; Open PDF.
• Haag, Rudolf, and Daniel Kastler. “An Algebraic Approach to Quantum Field Theory.” Journal of Mathematical Physics 5 (1964): 848–861. DOI.