An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G –> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo units of K. If one substitutes “endomorphism” for “automorphism” in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase “inner endomorphism” in the literature, some overlapping the one introduced here, are noted; the concept of an inner {\em derivation} of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a “co-inner” endomorphism is briefly examined. Several questions are posed.

💡 Research Summary

The paper revisits the classical notion of an inner automorphism of a group or an associative algebra from a purely categorical standpoint, eliminating any reference to individual elements. In the first part the author shows that an automorphism φ of a group G is “inner” precisely when, for every group H and every homomorphism f : G → H, there exists a unique automorphism Φ_H of H making the diagram commute (i.e., Φ_H ∘ f = f ∘ φ). This functorial extension property characterises inner automorphisms without mentioning conjugation. The set of all such functorial systems of automorphisms is naturally isomorphic to G itself; the isomorphism sends g∈G to the family of conjugations by g in every target group.

The same idea is then transferred to an associative algebra R over a field K. An automorphism of R that extends functorially to every K‑algebra S along any homomorphism R→S is exactly the class of automorphisms induced by multiplication with a unit of R, modulo the scalar units of K. Consequently the group of functorial automorphism systems is isomorphic to R^×/K^×. This recovers the familiar description of inner automorphisms of an algebra while emphasizing the role of the base field’s units.

When the word “automorphism’’ is replaced by “endomorphism’’ the situation diverges dramatically. For groups the only extra functorial endomorphisms are the trivial one (sending every element to the identity). No non‑trivial inner endomorphisms exist beyond the usual inner automorphisms. By contrast, for K‑algebras a new class of “inner endomorphisms’’ appears. These are precisely the endomorphisms arising from the multiplier algebra M(R) or, equivalently, from centralizing operators familiar to functional analysts (e.g., left multiplication by an element of the double centralizer). Such maps need not be invertible, yet they satisfy the same universal extension property: given any K‑algebra S and any homomorphism f:R→S, there is a unique endomorphism Ψ_S of S making the diagram commute. This construction is largely unknown to classical ring theorists but is standard in the theory of C∗‑algebras and Banach algebras.

The author proceeds to explore analogous notions in several other categories. In the category of left R‑modules, inner endomorphisms correspond to multiplication by central elements of R; in the arrow category (objects are morphisms) the definition yields a “double‑inner’’ condition. For Lie algebras the functorial inner derivations turn out to be exactly those derivations given by the adjoint action of a central element, linking the categorical viewpoint with the classical theory of inner derivations.

A dual concept, termed “co‑inner’’ endomorphism, is introduced. Instead of requiring a family of endomorphisms on all targets extending a given map, one demands a family on all sources that co‑extend a given map. This duality mirrors the usual categorical opposite and leads to a brief discussion of co‑inner maps in comodule and coalgebra settings.

The paper concludes with a list of open problems: (1) In which categories does the functorial extension property force every inner endomorphism to be an automorphism? (2) How does the structure of the multiplier algebra control the classification of inner endomorphisms for non‑unital algebras? (3) Can co‑inner endomorphisms be employed to obtain new invariants in representation theory or homological algebra? (4) What are the relationships between the categorical inner‑automorphism group and the usual inner‑automorphism group in higher‑categorical contexts? These questions point toward a rich interplay between categorical algebra, classical ring theory, and functional analysis, suggesting many avenues for future research.