Local spin matrix algebras and their factor inclusions

Fix an integer dimension $d\ge 1$, the countable site set $\mathrm{\Gamma }={\mathbb{Z}}^d$ of Countability of every finite-dimensional integer lattice, and one finite spin dimension $q\ge 1$, where $q$ is a positive integer (so $q\in \mathbb{N}$ and $1\le q$, by The natural numbers $\mathbb{N}$ (von Neumann) and Order on the natural numbers). For $n\in \mathbb{Z}$, define $\mathrm{\mid }n\mathrm{\mid }=n$ if $0\le n$ and $\mathrm{\mid }n\mathrm{\mid }=-n$ if $n<0$, using the integer order and additive inverse of Order on the integers and Arithmetic on the integers. Totality and compatibility of the integer order with addition (The integers form a totally ordered ring) give $0\le \mathrm{\mid }n\mathrm{\mid }$ and $-\mathrm{\mid }n\mathrm{\mid }\le n\le \mathrm{\mid }n\mathrm{\mid }$. Adding these bounds for $m,n\in \mathbb{Z}$ gives $\mathrm{\mid }m+n\mathrm{\mid }\le \mathrm{\mid }m\mathrm{\mid }+\mathrm{\mid }n\mathrm{\mid }$. Equip $\mathrm{\Gamma }$ with the max-coordinate metric $d_{\mathrm{\infty }}(x,y)={\max }_{1\le r\le d}\mathrm{\mid }{x}_r-y_r\mathrm{\mid }$; the maximum exists because the coordinate set is finite and nonempty, symmetry and separation hold coordinatewise, and the absolute-value triangle inequality gives $\mathrm{\mid }{x}_r-z_r\mathrm{\mid }\le \mathrm{\mid }{x}_r-y_r\mathrm{\mid }+\mathrm{\mid }{y}_r-z_r\mathrm{\mid }$ for every coordinate, hence the triangle inequality for $d_{\mathrm{\infty }}$. A finite region is a finite subset $\mathrm{\Lambda }\subseteq \mathrm{\Gamma }$; its cardinality is $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$ (The cardinality $\mid A\mid$ of a finite set).

For a finite region $\mathrm{\Lambda }$, let $S_{\mathrm{\Lambda }}=\{1,\dots ,q{\}}^{\mathrm{\Lambda }}$ be the set of spin configurations on $\mathrm{\Lambda }$. Induction on $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$ gives $\mathrm{\mid }{S}_{\mathrm{\Lambda }}\mathrm{\mid }=q^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}$: the empty region has one configuration, and for $x\in \mathrm{\Lambda }$ the restriction map $s\mapsto (s{\mathrm{\mid }}_{\mathrm{\Lambda }\setminus \{x\}},s(x))$ is a bijection $S_{\mathrm{\Lambda }}\to S_{\mathrm{\Lambda }\setminus \{x\}}\times \{1,\dots ,q\}$, so adjoining one site multiplies the number of configurations by $q$, according to the recursion defining natural-number powers in Exponentiation of natural numbers, $m^n$, and its agreement with the integer power in $\mathbb{R}$. Let ${\mathcal{H}}_{\mathrm{\Lambda }}$ be the vector space of functions $f:S_{\mathrm{\Lambda }}\to \mathbb{C}$ with inner product $⟨f,g⟩={\sum }_{s\in S_{\mathrm{\Lambda }}}f(s)\overline{g(s)}$. For each $s\in S_{\mathrm{\Lambda }}$, let $e_s$ be the indicator function of $\{s\}$; these vectors form an orthonormal basis. For $\mathrm{\Lambda }=\mathrm{\varnothing }$, $S_{\mathrm{\Lambda }}$ is a singleton and ${\mathcal{H}}_{\mathrm{\Lambda }}\cong \mathbb{C}$. The local spin matrix algebra is

${\mathcal{A}}_{\mathrm{\Lambda }}:=\mathcal{B}({\mathcal{H}}_{\mathrm{\Lambda }})\cong M_{q^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}}(\mathbb{C}),$

Here $\mathcal{B}({\mathcal{H}}_{\mathrm{\Lambda }})$ denotes all complex-linear endomorphisms of the finite-dimensional Hilbert space; they are bounded. The algebra has composition, Hilbert adjoint, identity $I_{\mathrm{\Lambda }}$, and induced operator norm

$\mathrm{\parallel }T\mathrm{\parallel }:={\sup }_{\mathrm{\parallel }\xi \mathrm{\parallel }=1}\mathrm{\parallel }T\xi \mathrm{\parallel }\mathrm{.}$

The finite local trace is ${\mathrm{Tr}}_{\mathrm{\Lambda }}(T):={\sum }_{s\in S_{\mathrm{\Lambda }}}⟨T{e}_s,e_s⟩$ in the displayed configuration basis $(e_s)$. This is the usual finite-matrix trace; basis independence and its cyclic/unitary identities are supplied by Finite matrix spectral and state calculus.

If $\mathrm{\Lambda }\subseteq {\mathrm{\Lambda }}^{\mathrm{\prime }}$, restriction of configurations gives the canonical factorization ${\mathcal{H}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}\cong {\mathcal{H}}_{\mathrm{\Lambda }}\otimes {\mathcal{H}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }}$, with sites ordered lexicographically. The factor inclusion is the unital $\ast$-homomorphism ${\iota }_{\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{\prime }}}:{\mathcal{A}}_{\mathrm{\Lambda }}\to {\mathcal{A}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}}$ given by $T\mapsto T\otimes I_{{\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }}$. A localization region for $T$ is any finite $\mathrm{\Lambda }$ for which $T\in {\mathcal{A}}_{\mathrm{\Lambda }}$ under these inclusions; a localization region need not be minimal.

The ordinary operator norm makes each local matrix algebra a nonzero unital complete C*-algebra, and every positive normalized complex-linear functional is uniquely a trace pairing with a density matrix, by Finite matrix C-star algebra and density-state duality. This includes the empty region, whose Hilbert space is $\mathbb{C}$. This definition does not posit an infinite-dimensional tensor-product Hilbert space.