Nonunital

Nonunital is a term used in algebra and analysis to describe structures that do not contain a multiplicative identity element. In rings, algebras, and related objects, a unital (or unitary) structure has a distinguished element 1 such that 1·a = a·1 = a for all elements a. If such an element is absent, the structure is called nonunital. This distinction is common in associative algebras over a field, C-algebras, and Banach algebras, where the lack of a unit affects both theory and applications.

Nonunital algebras often arise as ideals within larger unital algebras or as natural objects in functional-analytic contexts, such as algebras of operators that do not include the identity operator. For example, the ideal of compact operators K(H) on an infinite-dimensional Hilbert space H is a nonunital C-algebra, while B(H), the algebra of all bounded operators, is unital.

A standard tool for dealing with nonunital algebras is unitization, which adjoins a unit to obtain a unital algebra. For a nonunital algebra A over a field, the unitization A~ can be formed as A ⊕ F with multiplication (a, λ)(b, μ) = (ab + λb + μa, λμ). The embedding a ↦ (a, 0) identifies A with a two-sided ideal of A~. This construction provides a universal way to extend homomorphisms from A to unital targets.

In the C-algebra setting, many results involve approximate identities—nets (e_i) in A with ||ae_i − a|| → 0 and ||e_i a − a|| → 0 for all a in A. The multiplier algebra M(A) of a nonunital C-algebra A is the largest unital C-algebra in which A sits as an essential ideal; for K(H), M(K(H)) ≅ B(H).

Morphisms between nonunital algebras preserve addition and multiplication, but need not map identities, since none exist.