Finite-Mode Complex CAR Algebra

Fix an integer $n\ge 0$. Let $F_n$ be the complex vector space with basis all finite words in the symbols $a_1,\dots ,a_n,a_1^{†},\dots ,a_n^{†}$, including the empty word $1$. Give $F_n$ concatenation as multiplication and the conjugate-linear involution that reverses each word and exchanges $a_j$ with $a_j^{†}$. Thus $F_n$ is the free unital complex $\ast$-algebra on $a_1,\dots ,a_n$.

For $x,y$ in an algebra write $\{x,y\}=xy+yx$. Let $J_n$ be the smallest two-sided $\ast$-ideal in $F_n$ containing

$\{a_i,a_j\},\{a_i^{†},a_j^{†}\},\{a_i,a_j^{†}\}-{\delta }_{ij}1(1\le i,j\le n)\mathrm{.}$

Here a two-sided $\ast$-ideal is a complex linear subspace closed under left and right multiplication and under $\ast$. It exists as the intersection of all such ideals containing these relations; the collection is nonempty because it contains $F_n$. The relations are $\ast$-stable, since taking adjoints exchanges the first two families and changes the third relation with indices $i,j$ to the same relation with indices $j,i$. Define the algebraic finite-mode CAR $\ast$-algebra by ${\mathsf{C}\mathsf{A}\mathsf{R}}_n^{\mathrm{a}\mathrm{l}\mathrm{g}}=F_n\mathrm{/}{J}_n$, with generators denoted by the same symbols. Its quotient map gives the universal property: every tuple in a unital complex $\ast$-algebra satisfying these relations induces a unique unital $\ast$-homomorphism from ${\mathsf{C}\mathsf{A}\mathsf{R}}_n^{\mathrm{a}\mathrm{l}\mathrm{g}}$. For $n=0$ there are no generators or relations and ${\mathsf{C}\mathsf{A}\mathsf{R}}_0^{\mathrm{a}\mathrm{l}\mathrm{g}}=\mathbb{C}$.

This definition is algebraic; the finite-mode universal C*-completion is identified in The Finite-Mode CAR Algebra Is a Full Matrix Algebra.