CAR C*-Algebra over a Complex Hilbert Space

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for this norm-completion construction, and let $H$ be a complex Hilbert space. Order its finite-dimensional complex subspaces by inclusion. This is a directed system because the span $F+G$ of any two finite-dimensional subspaces is finite-dimensional. To each such $F$, assign the finite-dimensional CAR C*-algebra of Finite-Dimensional CAR over an Inner-Product Space Is a Matrix Algebra, with the generator-preserving maps of Finite-Subspace CAR Inclusions Are Isometric. Identify each stage with its image under these injective maps and let ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(H)$ be their algebraic union.

The common finite-stage norm is well defined on this union because every inclusion is isometric. Under Countable Choice, Every metric space has a completion, constructed as the equivalence classes of its Cauchy sequences supplies a metric completion and The metric completion of a normed space carries a unique compatible Banach-space structure equips it with the compatible Banach-space structure; this is the norm completion of Completion of a normed space. Define the CAR C-algebra over $H$**, $\mathsf{C}\mathsf{A}\mathsf{R}(H)$, to be this norm completion of ${\mathsf{C}\mathsf{A}\mathsf{R}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(H)$. Addition, multiplication, scalar multiplication, and involution are the compatible finite-stage operations; submultiplicativity and isometry of the involution make them extend continuously to the completion, and the C-identity passes to limits. The identity is nonzero because the zero-dimensional stage is $\mathbb{C}$ and every inclusion preserves its unit. For $f\in H$, write $c(f)$ for the image of the generator in the stage $\mathrm{span}\{f\}$; the universal inductive-limit property and its norm are proved in CAR over a Hilbert Space Has the Universal Inductive-Limit Norm.