Theorem: Observable sequence-limit construction

The following statements hold.

1. Each ${\mathfrak{A}}_{\infty }(O)$ is a unital $C^{\ast }$-subalgebra of ${\mathfrak{M}}_{\infty }$, and isotony passes to the limit.

2. If the two scale algebras commute for all sufficiently large $n$, their limit local algebras commute. More generally, the same conclusion follows if every pair of bounded local representative sequences has commutator norm tending to zero.

3. UCP maps $T_n:{\mathfrak{M}}_{\mathrm{out},n}\to {\mathfrak{M}}_{\mathrm{in},n}$ induce a UCP map

   $$T_{\infty }([a_n])=[T_n(a_n)].$$

4. If

   $$T_n({\mathfrak{A}}_{\mathrm{out},n}(Y))\subseteq {\mathfrak{A}}_{\mathrm{in},n}(I)$$

   for all sufficiently large $n$, the induced map has the same containment and the tensor-free comparison theorem applies.

5. Complete recorded instruments with fixed finite alphabets, independent matrix references, and matched equalities of device actions pass to this quotient.

Proof. The local bounded product is a $C^{\ast }$-algebra. Its intersection with the ambient null ideal is its own norm-null ideal. Formula (eq:quotientnorm) shows that its quotient embeds isometrically into the ambient quotient. Its image is therefore closed, proving (i).

For (ii),

and the stated commutator sequences are norm null.

UCP maps are contractions at every matrix size. They send bounded sequences to bounded sequences and norm-null sequences to norm-null sequences. The coordinatewise product map is UCP, and quotienting gives a UCP map, proving (iii).

For (iv), discard finitely many terms and use the indicated local representatives. Equivalently, use their local UCP restrictions to construct the induced factorization. The resulting local image is in (eq:limitlocalalgebra).

Finite direct sums and finite matrix amplifications commute with this construction. Equality on the specified local sequence algebra passes term by term or after a norm-null error, proving (v).