Quasi-Unitality in Algebra & Category Theory

• Quasi-unitality conditions are defined as structural properties that extend nonunital objects to frameworks exhibiting unit-like behavior.
• They utilize criteria such as extended Hilbert bounds, t-unitality, and homotopy-coherent quasi-units to bridge algebraic, analytic, and categorical settings.
• These conditions enable functional extensions, module category equivalences, and the construction of local units in operator algebras and ∞-categories.

Quasi-unitality conditions describe a spectrum of structural properties in algebra, analysis, and higher category theory, formalizing the presence of “unit-like” behavior in settings lacking strict two-sided units. These conditions provide intrinsic or external criteria for extending non-unital or partially unital objects—rings, algebras, categories, operator systems—to frameworks that admit well-behaved unitizations, completions, or local unital invariants. Quasi-unitality plays a technically decisive role in the extension of functionals, the structure theory of modules, the homotopy theory of ∞-categories, the duality of quantales, and the classification of operator algebra multipliers.

1. Quasi-unitality in Algebraic and Categorical Structures

Quasi-unitality arises in several algebraic contexts as a relaxation or internalization of the classical notion of unitality:

• Quasi-Unitals in Quasi *-Algebras: In the context of quasi *-algebras, quasi-unitality is encoded by analytical bounds ensuring the extension of a Hermitian linear functional φ: A → ℂ, defined on a quasi *-algebra $(A, A_0)$ without unit, to a representable positive linear functional on its unitization (Bellomonte, 2013). The explicit “extended Hilbert bound” (EHB) condition ensures the compatibility of φ with the adjoining of a unit to $(A, A_0)$.
• Module Theory over Nonunital Rings: In the tensor–Hom formalism, t-unitality for modules (the isomorphism $R \otimes_R M \xrightarrow{\sim} M$) and c-unitality (the canonical map $P \xrightarrow{\sim} \operatorname{Hom}_R(R, P)$) furnish two invariants of quasi-unitality for modules and bimodules over non-unital rings (Positselski, 2023). These properties are mutually reflective under a general quotient equivalence with Serre subcategories of null modules, capturing the essence of “weakly unital” module categories.
• Weak Unitality in ∞-Categories and Segal Objects: Non-unital ∞-categories (modeled by semiSegal or semi-simplicial spaces) are quasi-unital if every object admits a weakly invertible, homotopy-coherent self-morphism—an idempotent equivalence or quasi-unit (Harpaz, 2012, Oldervoll, 16 Jan 2026). Localizations at marked horns or “outer degeneracies” recast the presence of such units as local horn-filling or marking conditions.

2. Comparative Frameworks: Categorical, Analytic, and Algebraic

Several foundational frameworks illustrate the shape and consequences of quasi-unitality:

Context	Quasi-Unitality Notion	Key Criterion
Quasi *-algebras (Bellomonte, 2013)	Extended Hilbert bound (EHB)	$|\phi(x^*)| \leq C \, P_x(\phi)^{1/2}$
Nonunital Rings (Positselski, 2023)	t-unitality, c-unitality, s-unitality	$\mu_M: R \otimes_R M \to M$ isomorphism
∞-Categories (Harpaz, 2012, Oldervoll, 16 Jan 2026)	Existence of homotopy units	Every object admits idempotent (invertible) end
Operator Algebras (Brown, 2018)	Local unitality of ideals	Each ideal $I_j$ with local unit $u_j$
Quantales (Lacroix et al., 2022)	Local (quasi-)units $u = \wedge_{x}(x\setminus x)$	$x * u = x$, $u * x = x$ iff $u \geq x\setminus x$

These formulations are not mutually reducible but exhibit thematic parallels: quasi-unitality always references the internal sufficiency for absorbing, extending, or reconstructing a unit action from substructure or local invariants.

3. Characteristic Results and Main Theorems

Specific theorems reveal the technical import of quasi-unitality:

• Extension of Functionals (Quasi *-Algebras): Bellomonte’s main result [(Bellomonte, 2013), Proposition 3.5] states that a Hermitian linear functional φ on $(A, A_0)$ is extendable to a *-representable functional on the unitization if and only if (EHB) holds:

$$|\phi(x^*)| \leq C \left[ \sup \{|\phi(x^*a)|^2 : a \in A_0,\ \phi(a^* a) = 1 \} \right]^{1/2}.$$

This strictly generalizes the classical Hilbert-boundedness criterion on *-algebras.

• Monoidal and Abelian Equivalence for Modules: Over a t-unital ring, the abelian categories of t-unital and c-unital modules are equivalent, and both correspond to the Serre quotient of all modules by null-modules (modules with zero $R$-action) (Positselski, 2023).
• Unicity of Unit Completion in ∞-Categories: Harpaz proves that for any quasi-unital semiSegal space, the unital and complete Segal structure is determined uniquely up to contractible choice, once quasi-units are specified [(Harpaz, 2012), Thm 3.3.1]. Oldervoll extends and unifies these perspectives for inner Kan and marked inner Kan models (Oldervoll, 16 Jan 2026).
• When Quasi-Multiplier Equals Multiplier: For $\sigma$-unital $C^*$-algebras, every quasi-multiplier is a multiplier precisely when the algebra decomposes as the direct sum of a dual $C^*$-algebra and a locally unital $C^*$-algebra (Brown, 2018). Local unitality is characterized via local units assigned to dense families of ideals.

4. Constructions: Unitification, Local and Phase Quotients

Quasi-unitality conditions frequently underpin explicit unitification constructions or the analysis of local/phase quotients:

• Unitification of Weakly p.q.-Baer *-Rings: A *-ring is p.q.-Baer if and only if it is weakly p.q.-Baer and unital (Khairnar et al., 2016). Explicit construction: for a weakly p.q.-Baer *-ring $R$ and suitable $K$, form $R^+ = R \oplus K$ with induced multiplication and involution; $R^+$ is a p.q.-Baer ring with preserved central covers.
• Phase Quantale Construction: In unitless Frobenius quantales, passage to the fixed points of a quantic nucleus $j$ yields a phase quantale $Q_j$ with a true unit, representing a quotient through a Serre Galois connection (Lacroix et al., 2022). Local quasi-units $u = \wedge_x(x \setminus x)$ exist even when no global unit does.

5. Interplay With Homotopical and Horn-Filling Conditions

In higher category theory, quasi-unitality is often formalized via horn-filling conditions:

• Inner Horns and Markings: The existence of idempotent equivalences in semi-simplicial or semiSegal models is equated with the ability to fill inner horns up to homotopy and to mark specific arrows as invertible (Harpaz, 2012, Oldervoll, 16 Jan 2026).
• 2-Segal Spaces: Every 2-Segal space admits a unital structure; in effect, the very weak "quasi-unit" data implicit in the 2-Segal condition can be upgraded to full unitality via pullback/retract arguments (Feller et al., 2019). This demonstrates that for certain “quasi-unital” combinatorics, no obstruction remains to global unitalization.

6. Examples and Illustrative Cases

Concrete examples clarify the reach and boundaries of quasi-unitality:

• Successful Extension (Bellomonte, 2013): For $$A = \{ f \in C(\mathbb{R}) : \int |f(x)| e^{-x^2} dx < \infty \}$$, $A_0 = C_c(\mathbb{R})$, the evaluation functional $f \mapsto f(0)$ satisfies the EHB, thus admits a unitization.
• Module-theoretic Equivalences (Positselski, 2023): For s-unital rings (in Tominaga's sense), every s-unital module is t-unital and vice versa, guaranteeing the equivalence of several “weakly unital” module categories.
• Quantales (Lacroix et al., 2022): In the Raney tight-endomap quantale, the local quasi-unit exists but is not a global unit unless the base lattice is completely distributive.
• $C^*$-Algebras (Brown, 2018): $A = C_0(X) \otimes M_n(\mathbb{C})$ (for $X$ locally compact Hausdorff, $M_n$ unital) is locally unital and thus quasi-unital in the operator algebraic sense.

7. Implications, Equivalences, and Non-Extendibility Results

Quasi-unitality provides a template for:

• Extensibility and Uniqueness Criteria: Many settings admit necessary and sufficient quasi-unital conditions for the extension or reconstruction of strictly unital structures.
• Obstructions to Global Unitization: For certain quantales, no embedding into a strictly unital quantale can preserve primitive negation operations unless the local quasi-unit is already global (Lacroix et al., 2022).
• Module Category Equivalences and Closure Properties: Abelian and monoidal equivalence of weakly unital module categories hinges on t-unitality and c-unitality, as in the tensor–Hom theory (Positselski, 2023).

A plausible implication is that quasi-unitality conditions not only extend the reach of unital techniques into nonunital or locally unital settings but can also serve as a sharp obstruction to unitality-preserving extensions or functorial constructions, particularly in duality-sensitive frameworks.

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References:

• (Bellomonte, 2013) G. Bellomonte, Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras
• (Positselski, 2023) L. Positselski, Tensor–Hom formalism for modules over nonunital rings
• (Harpaz, 2012) Y. Harpaz, Quasi-unital $\infty$-Categories
• (Oldervoll, 16 Jan 2026) P. Oldervoll, Quasi-unitial Inner Kan Spaces
• (Lacroix et al., 2022) D. de Lacroix, F. Santocanale, Unitless Frobenius quantales
• (Brown, 2018) L.G. Brown, When is Every Quasi-Multiplier a Multiplier?
• (Feller et al., 2019) M. Dyckerhoff, T. Kapranov, Every 2-Segal space is unital
• (Khairnar et al., 2016) D. Khairnar, B. Waphare, Unitification of Weakly p.q.-Baer *-Rings