Locally m-Convex Topological Algebras

• Locally m-convex topological algebras are defined by submultiplicative seminorms, ensuring joint continuity of multiplication.
• They underpin representation theorems, measure-theoretic functionals, and functional calculi across operator theory and quantum field theory.
• Their structured topology supports automatic continuity of multiplicative functionals and enables extensions to noncommutative and generalized function settings.

A locally $m$-convex topological algebra is a topological algebra whose topology is defined by a family of submultiplicative seminorms. These algebras form a foundational subclass of locally convex topological algebras, underpinning critical analytic, algebraic, and structural results across real algebraic geometry, operator algebras, stochastic analysis, and quantum field theory. Their defining $m$-convexity property manifests as joint continuity for multiplication, enabling the development of deep representation theorems, measure-theoretic functionals, and functional calculi.

1. Definition, Axioms, and Basic Properties

A locally $m$-convex algebra is a topological algebra $A$ over $\mathbb{R}$ or $\mathbb{C}$ admitting a neighborhood basis at $0$ consisting of multiplicative sets, equivalently a topology generated by a family $(p_\lambda)_{\lambda \in \Lambda}$ of submultiplicative seminorms:

$$p_\lambda(xy) \leq p_\lambda(x) p_\lambda(y) \quad \forall x, y \in A.$$

Every Hausdorff locally convex topology defined by such a saturated family is locally multiplicatively convex ("lmc"). The completed space with respect to this topology retains a rich algebraic structure compatible with analysis, including the existence of a Gelfand spectrum and the potential for uniform and saturated structures (Ghasemi et al., 2012, Azhari, 2013, Azhari, 2013).

Key features include:

• Joint continuity of multiplication in the topology.
• Existence of subalgebras and quotient algebras which are normed or Banach algebras under induced seminorms.
• Automatic continuity of multiplicative functionals under weak regularity or functionally continuous assumptions.

2. Seminorms, Topological Structure, and Saturation

Submultiplicative seminorms not only define the topology but also mediate algebraic objects—functionals, cones, modules—and their closure. Saturated locally $m$-convex algebras satisfy stringent optimizing conditions:

• For any $p_\lambda(x) = 1$, there exists a multiplicative linear functional $f$ attaining $|f(x)|=1$ (Azhari, 2013).
• Uniform topological algebras arise when the seminorms further satisfy the square property $p(x^2) = p(x)^2$; in the $m$-convex case, saturation, uniformity, and "Cochran algebra" property are equivalent structures (Azhari, 2013).

The Gelfand map $G: A \to C(M)$, with $M$ the space of continuous nonzero multiplicative linear functionals, realizes these algebras as concrete function spaces when the topology is induced by uniform seminorms.

3. Representation Theorems and Measure-Theoretic Functionals

Jacobi’s Representation Theorem asserts that positive linear functionals continuous with respect to an $m$-convex topology are exactly those representable as integrals against unique Radon measures supported on compact subsets of the Gelfand spectrum:

$L(a) = \int_{X(A)} \hat{a}(\alpha) \, d\mu(\alpha)$

for $L: A \rightarrow \mathbb{R}$ with $L(S) \subset [0, \infty)$ and $S$ a $\sum A^{2d}$-module (Ghasemi et al., 2012). The support of $\mu$ is tightly controlled by positivity sets and submultiplicative seminorms, establishing a bridge between topology, order structure, and measure theory.

In symmetric algebras over locally convex spaces, extensions of these representation theorems guarantee every continuous, positive functional is representable as a Radon measure on closed balls $B_1(i p')$ in the algebraic dual $V^*$, sharpening classical moment problem resolutions and opening avenues into infinite-dimensional analysis (Ghasemi et al., 2015).

4. Functional Calculus, Operator Theory, and Noncommutative Extensions

Locally $m$-convex algebras integrate seamlessly with functional calculus and operator theory:

• For locally convex quasi $C^*$-normed algebras, functional calculus for quasi-positive elements generalizes the $C^*$-algebra case via submultiplicative regularity conditions (Bagarello et al., 2012).
• Operator representations on dense domains in Hilbert spaces allow the transference of algebraic operations to linear or partial *-algebras.
• Noncommutative generalizations, including locally $m$-convex $C^*$-algebras and Banach algebra settings, underpin solution theory for singular stochastic PDEs and quantum statistical models.

In noncommutative probability, as with $q$-Gaussian processes ($q \in (-1,1)$), typical operator norms do not suffice to control intricate products and renormalisation insertions. The development of new submultiplicative Banach algebra norms such as

$\vvvert A \vvvert := \sum_{k=0}^n (k+1) \, C_q^{3/2} \, D_q^k \, \|F_k\|_0$

(with $A = \sum_{k=0}^n \xi_q^\circ(F_k)$) provides robust control in Wiener chaos expansions, enabling renormalised analytic estimates critical for noncommutative regularity structures, where the full machinery of modelled distributions and reconstruction theorems is lifted to the locally $m$-convex topological algebra setting (Chandra et al., 9 Sep 2025).

5. Examples and Applications

Locally $m$-convex algebras arise across a spectrum of mathematical models:

• Polynomial algebras $\mathbb{R}[X]$, quadratic modules, and cones underpin the solution of the multidimensional moment problem via submultiplicative seminorms tied to archimedean properties (Ghasemi, 2014).
• Subalgebras of nonlinear generalized functions, as in Colombeau theory, structured as inductive limits of Banach spaces of holomorphic functions, benefit from nuclearity and compactness, approximating locally $m$-convex behavior and facilitating embeddings of distributions and singular functions (Aragona et al., 2014).
• Realizations by linear vector fields and embeddings into Cuntz algebras leverage biorthogonal systems, expanding locally $m$-convex topological algebras into the operator algebra landscape, with further ramifications for infinite-dimensional Lie algebra representations (Bock et al., 2019).

6. Comparative Structural Insights and Generalizations

The locally $m$-convex condition is substantially more rigid and analytically powerful than general locally convex topologies:

• Joint continuity underpins automatic continuity results, uniqueness of uniform norms, and algebraic isomorphisms with function spaces.
• Saturated, uniform, and Cochran algebra structures converge in the $m$-convex case, confirming the robustness and utility of this setting for functional representation.
• Generalization to non-Archimedean modules proceeds via inductive limits over Archimedean enlargements, maintaining locally $m$-convex topology by direct limit construction (Ghasemi, 2014).

7. Open Directions and Research Frontiers

Recent advances motivate further exploration of locally $m$-convex topological algebras in:

• Extension of differential calculi and nonlinear operations on subalgebras of generalized functions, with potential connections to quantum field theory renormalization frameworks (Aragona et al., 2014).
• Structure theory and automatic continuity questions in infinite-dimensional settings, with applications in stochastic quantisation and matrix-valued SPDEs (Chandra et al., 9 Sep 2025).
• Deeper analysis of saturation and uniformity in topological algebras, refining Gelfand representations, and exploring functionally analytic properties in diverse contexts (Azhari, 2013, Azhari, 2013).

These developments reinforce the centrality of locally $m$-convex topological algebras as a foundational class, integrating algebraic positivity, analytic structure, and functional representation across pure and applied mathematics.