Unital Locally Matrix Algebras

• Unital locally matrix algebras are associative F-algebras in which every finite subset is contained in a unital subalgebra isomorphic to a full matrix algebra, ensuring local simplicity.
• They can be constructed as direct limits of finite-dimensional matrix algebras, linking the combinatorics of matrix sizes with algebraic invariants like the Steinitz number.
• Their rich structure underpins classification theorems, Morita equivalence, and connections to preserver problems and Lie-theoretic properties.

A unital locally matrix algebra is an associative unital algebra over a ground field $F$ in which every finite subset is contained in some unital subalgebra isomorphic to a full matrix algebra $M_n(F)$. Such algebras generalize both simple matrix rings and their infinite analogs, encapsulating locally simple behavior within a potentially infinite or uncountable-dimensional global structure. Their internal architecture is governed by the combinatorics of finite matrix sizes, encoded via Steinitz (supernatural) numbers, leading to powerful classification theorems, connections to direct limits, spectral invariants, Morita theory, and rich extensions to preserver problems and Lie-theoretic properties.

1. Formal Definition and Fundamental Examples

Let $F$ be a field. A unital $F$-algebra $A$ is called a unital locally matrix algebra if for every finite subset $\{a_1, \ldots, a_s\} \subseteq A$ there exists a unital subalgebra $B \leq A$, with $1_B = 1_A$ and $B \cong M_n(F)$ for some $n \geq 1$, such that $\{a_1, \ldots, a_s\} \subseteq B$ (Bezushchak et al., 2019). This condition is equivalent to expressing $A$ as a direct limit of finite-dimensional matrix algebras along unital embeddings:

$A = \varinjlim_{i} M_{n_i}(F).$

Key sources of such algebras include:

• Infinite tensor products: $A = \bigotimes_{i} M_{n_i}(F)$ for a family $(n_i)$;
• Clifford algebras: For an infinite-dimensional nondegenerate quadratic space $(V,f)$ (with $\mathrm{char}\,F \neq 2$), $\mathrm{Cl}(V,f)$ is unital locally matrix (Bezushchak, 11 Jan 2026).

2. Steinitz Numbers and Invariants

A Steinitz number (“supernatural number”) is a formal product $T = \prod_{p \text{ prime}} p^{e_p}$, where $e_p \in \mathbb{N} \cup \{\infty\}$ (Bezushchak et al., 2019, Bezushchak et al., 2020). These form a complete lattice under divisibility and uniquely encode the possible sizes of full matrix subalgebras in $A$.

Given a unital locally matrix algebra $A$, set

$$D(A) = \{ n \in \mathbb{N}\mid A \text{ contains a unital copy of } M_n(F)\}$$

and define the Steinitz number as

$$\mathrm{st}(A) = \mathrm{lcm}_{\mathrm{Steinitz}}(D(A)) = \prod_p p^{\,\sup\{v_p(n): n \in D(A)\}}.$$

Basic properties:

• The Steinitz number is multiplicative under tensor product:

$$\mathrm{st}(A \otimes B) = \mathrm{st}(A) \cdot \mathrm{st}(B).$$

• For diagonal direct limits $A = \varinjlim_i M_{n_i}(F)$, $\mathrm{st}(A) = \mathrm{lcm}_i(n_i)$.

In the countable-dimensional case, $\mathrm{st}(A)$ is a complete isomorphism invariant (Bezushchak et al., 2019, Bezushchak et al., 2020).

3. Structure Theory and Classification

3.1 Isomorphism and Universal Theory

Countable dimension:

The Steinitz number is a complete invariant: $A \cong B$ if and only if $\mathrm{st}(A) = \mathrm{st}(B)$, for unital locally matrix algebras of countable dimension (Bezushchak et al., 2019). Universal Theory:

For arbitrary dimensions (possibly uncountable), $A$ and $B$ satisfy exactly the same universal first-order sentences (with $1$) if and only if $\mathrm{st}(A) = \mathrm{st}(B)$. However, for uncountable dimension, isomorphism may fail even with equal Steinitz numbers (Bezushchak et al., 2019).

Morita equivalence:

Two countable-dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected: there is some rational $q>0$ with $st(B) = q \cdot st(A)$, meaning that $st(A)$ and $st(B)$ differ by integer exponents in only finitely many primes (Bezushchak et al., 2020). For uncountable dimension and non-locally-finite Steinitz numbers, this fails; there exist non-Morita-equivalent algebras with the same Steinitz invariant.

3.2 Spectrum and Saturated Sets

For a unital locally matrix algebra $A$, the set of Steinitz numbers of its unital corners,

$$\mathrm{Spec}(A) = \{ \mathrm{st}(eAe) \mid e^2 = e \in A,\, e \neq 0 \},$$

is a saturated subset of the set of all Steinitz numbers. In the unital case, $\mathrm{Spec}(A)$ has a unique maximal element—namely, $\mathrm{st}(A)$—and must be of the form $\{1,2,\ldots,n\}$ or $S(r,s) = \{ (a/b)s \mid a \leq rb,\, b|s \}$ with $r \in \mathbb{Q},\, s$ infinite (Bezushchak, 2020).

Scenario	Spectrum $\mathrm{Spec}(A)$	Steinitz Number $\mathrm{st}(A)$
$M_n(F)$ (finite matrix alg.)	$\{1,2,\ldots,n\}$	$n$
Direct limit of $M_{2^k}(F)$	$S(r,2^\infty)$ for $r$ rational	$2^\infty$
Infinite tensor $\bigotimes M_{n_i}(F)$	$S(\infty, s)$	$s = \prod_p p^{e_p}$ (depending on $n_i$)

4. Concrete Examples and Non-classical Phenomena

• Clifford algebras: For infinite $V$ with nondegenerate $f$ (char $\neq 2$), $\mathrm{Cl}(V,f)$ is unital locally matrix with $D(\mathrm{Cl}(V,f)) = \{2^m\}$, hence $\mathrm{st} = 2^\infty$ (Bezushchak et al., 2019).
• Generalized Clifford algebras: $\mathrm{Cl}_g(\ell,I)$ (with $\ell$ invertible in $F$ and $I$ infinite) is locally matrix, and $\mathrm{st} = \ell^\infty$ (Bezushchak et al., 2019).
• Arbitrary Steinitz numbers: Any infinite Steinitz number arises as the invariant of some unital locally matrix algebra. For non-primary infinite Steinitz numbers, direct limit or tensor constructions produce algebras not isomorphic to infinite tensor products of matrix factors (Bezushchak et al., 2019).

Non-primary decomposable examples:

There exist unital locally matrix algebras of uncountable dimension with infinite locally finite Steinitz number that do not admit a primary decomposition (i.e., a tensor product of primary locally matrix factors). The construction uses transfinite induction and Kurosh's embedding theorem to build such algebras (Bezushchak et al., 2019). If $A$ contains a countable subset whose centralizer in $A$ is $F \cdot 1$, it is not isomorphic to any infinite tensor product of finite matrix algebras.

5. Extensions, Direct Limits, and Regularity

Closure under extensions:

If $0 \to A \to B \to C \to 0$ is a short exact sequence of unital $K$-algebras with $A$ and $C$ locally matricial (unital locally matrix) algebras, then $B$ is also locally matricial, regardless of the base field (Goodearl, 17 Apr 2025).

Locally matricial algebras coincide with von Neumann regular direct limits of finite-dimensional simple matrix algebras. Each is unit-regular and locally unital, and every finite subset sits inside a finite-dimensional full matrix algebra (Goodearl, 17 Apr 2025). In countable dimension, these yield the (algebraic) AF algebras; the extension theorem is the algebraic analog of the Elliott AF-extension result.

Direct limits and classification:

Every unital locally matrix algebra can be written as a direct limit $\varinjlim M_{n_i}(F)$. In countable dimension, isomorphism is determined by the Steinitz lcm of $\{n_i\}$, and embeddings (as approximative corners) are controlled by inclusion of spectra (Bezushchak, 2020).

6. Preserver Problems and Lie Structure

Normalized rank and determinant preservers:

A linear map $\varphi: A \to B$ between unital locally matrix algebras preserving normalized rank must globally decompose (up to inner automorphisms) as a direct sum of a homomorphism and antihomomorphism, defined by orthogonal idempotents (Bezushchak, 11 Jan 2026). For surjective $\varphi$, this reduces to (anti-)isomorphisms up to inner twists. Deteminant preservers (over $\mathbb{R}$ or $\mathbb{C}$) also have this structural form.

Map Type	Structure	Additional Condition
Normalized-rank-preserving, surjective	$X\psi(a)Y$, $\psi$ (anti-)isomorphism	None (simplicity of $B$)
Normalized-determinant-preserving	$X\psi(a)Y$ as above	$\mathrm{ndet}(XY)=1$

Lie structure:

• The Lie algebra $A^{(-)}/F\cdot 1$ is simple if and only if either $\mathrm{char}\,F=0$, or for each $p>0$, the $p$-adic exponent of $\mathrm{st}(A)$ is $0$ or $\infty$ (Bezushchak, 2020).
• The Lie algebra of derivations $\mathrm{Der}(A)$ is topologically simple under the same conditions.
• Example: For $A$ a direct limit of $M_{2^k}(F)$ (char $F \neq 2$), $A^{(-)}/F$ and $\mathrm{Der}(A)$ are both simple.

7. Open Problems and Further Directions

• Locally finite Steinitz numbers: It is conjectured that if $\mathrm{st}(A)$ is locally finite (i.e., all $e_p<\infty$), then $\dim_F A$ must be countable (Bezushchak et al., 2019).
• Primary decomposition: Not all unital locally matrix algebras admit a (possibly infinite) tensor product of primary subalgebras; stronger criteria are needed in uncountable dimensions (Bezushchak et al., 2019).
• Preserver theory: Questions remain about characterizing linear maps preserving other normalized invariants (e.g., spectral data), the behavior in characteristic $2$, and nonunital or topological generalizations (Bezushchak, 11 Jan 2026).
• Connections to infinite operator algebras: The algebraic theory parallels the classification of UHF $C^*$-algebras via supernatural numbers; Bratteli–Elliott invariants and $K_0$-theory play a key role (Bezushchak et al., 2019, Goodearl, 17 Apr 2025).
• Lie-theoretic generalizations: Classification of diagonal locally simple Lie algebras uses Steinitz-like invariants (Bezushchak et al., 2019). The interplay between the associative and Lie structures continues to be explored.

In summary, unital locally matrix algebras are governed by their matrix subalgebra structure, completely parameterized (in the countable-dimensional case) by their Steinitz number. They serve as a structural bridge between finite matrix theory, infinite tensor products, direct limit algebras, and the study of large algebraic or operator-theoretic systems, with the Steinitz number emerging as the universal invariant for isomorphism, Morita equivalence, spectral theory, and many further structures.