Countably Infinite-Mode CAR C*-Algebra

Let $\mathbb{N}=\{1,2,\dots \}$ be the countable mode set. For each $n$ take the universal finite-mode C*-algebra ${\mathsf{C}\mathsf{A}\mathsf{R}}_n$ whose dense algebraic presentation is Finite-Mode Complex CAR Algebra. Identify each stage with its image in the next under the isometric inclusion of Canonical Finite-Mode CAR Inclusions Are Isometric. The inclusions are compatible, injective, and preserve the C*-norm. Their union ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}={\bigcup }_{n\ge 0}{\mathsf{C}\mathsf{A}\mathsf{R}}_n$ is therefore a normed $\ast$-algebra with norm $\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }x{\mathrm{\parallel }}_{{\mathsf{C}\mathsf{A}\mathsf{R}}_n}$ whenever $x\in {\mathsf{C}\mathsf{A}\mathsf{R}}_n$; isometry makes this independent of the stage. Define the countably infinite-mode CAR C-algebra* ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathbb{N}}$ to be the norm completion of ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ in the sense of C star algebra. Its canonical generator $a_j$ is the image of the $j$th finite-mode generator in any stage $n\ge j$. The C*-axioms pass to the completion because each finite stage satisfies them and multiplication and involution are continuous. The universal property of this inductive-limit construction is proved in The Countable CAR Algebra Has the Universal Inductive-Limit Norm.

The existence of this particular completion needs no choice principle. Let ${\pi }_n:{\mathsf{C}\mathsf{A}\mathsf{R}}_n\to M_{2^n}(\mathbb{C})$ be the occupation-basis isomorphism of The Finite-Mode CAR Algebra Is a Full Matrix Algebra, and let $D_n\subseteq {\mathsf{C}\mathsf{A}\mathsf{R}}_n$ be the inverse image of $M_{2^n}(\mathbb{Q}(i))$, where $\mathbb{Q}(i)=\{a+ib:a,b\in \mathbb{Q}\}$. Write ${\mathbb{N}}_0=\{0,1,2,\dots \}$ for coding indices, to distinguish it from the positive mode set $\mathbb{N}$. The theorem $\mathbb{Q}$ is countably infinite supplies a surjection ${\mathbb{N}}_0\to \mathbb{Q}$, and pairing two such indices by $\mathbb{N}\times \mathbb{N}\approx \mathbb{N}$ gives an explicit enumeration of $\mathbb{Q}(i)$. In these coordinates the canonical inclusion is $A\mapsto A\otimes I_2$ (Canonical Finite-Mode CAR Inclusions Are Isometric), so $D_n\subseteq D_{n+1}$. Each finite array of Gaussian rationals is coded by a finite tuple in ${\mathbb{N}}_0$; repeated pairing, followed by pairing with the stage index, enumerates $D={\bigcup }_{n\ge 0}{D}_n$ without choosing from a family of stage enumerations. Entrywise approximation of real and imaginary parts by rationals (The rationals embed densely in the reals) shows Gaussian-rational matrices are dense in each finite-dimensional matrix stage: in fixed size $m$, $\mathrm{\parallel }A-B{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le m{\max }_{r,s}\mathrm{\mid }{a}_{rs}-b_{rs}\mathrm{\mid }$. Hence $D$ is dense in ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$.

Concretely, form the set of Cauchy sequences in ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$, identify two sequences when the norm of their termwise difference tends to zero, and use the limiting norm and termwise algebra operations. Write $\widehat{d}([a_n],[b_n])={\lim }_n\mathrm{\parallel }{a}_n-b_n\mathrm{\parallel }$ for the resulting completion metric. Products are well defined because Cauchy sequences are bounded and $\mathrm{\parallel }{x}_m{y}_m-x_n{y}_n\mathrm{\parallel }\le \mathrm{\parallel }{x}_m\mathrm{\parallel }\mathrm{\parallel }{y}_m-y_n\mathrm{\parallel }+\mathrm{\parallel }{x}_m-x_n\mathrm{\parallel }\mathrm{\parallel }{y}_n\mathrm{\parallel }$; the involution is isometric. Thus the completion is a normed $\ast$-algebra, and its C*-identity passes to limits. It is complete without Countable Choice: the constant-sequence image of ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ is dense in the sequence-class completion, so $D$ is dense there too. Fix the explicit enumeration $q:{\mathbb{N}}_0\twoheadrightarrow D$. For any Cauchy sequence $(x_m)$ of completion points, let $d_m$ be the first $q(r)$ within $2^{-m}$ of $x_m$. The least-index rule defines each $d_m$ uniquely. The triangle inequality shows $(d_m)$ is Cauchy; for $m,n$ sufficiently large, $\widehat{d}(x_m,d_n)\le \widehat{d}(x_m,x_n)+2^{-n}$ is arbitrarily small, so the class $[d_n]$ is the limit of $(x_m)$. Hence this sequence-class construction is a completion of ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$ and defines ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathbb{N}}$ choice-free. Its canonical generators and inductive-limit norm are unchanged.