In finite dimensions, the determinant has a defining algebraic role that’s easy to forget: it is the universal abelianization map of the general linear group. Concretely, for (GL_n(\mathbb{C})), the kernel of the determinant is exactly the commutator subgroup. This is why the identity
[
\det(ABA^{-1}B^{-1}) = 1
]
holds for all invertible matrices (A,B).

In infinite dimensions, this role breaks down. Classical extensions of the determinant—Fredholm, regularized ((\det_p)), zeta, Fuglede–Kadison—are all scalar-valued and defined only on restricted domains. Outside special regimes, their kernels are strictly larger than the commutator subgroup. As a result, they cannot realize the abelianization of the infinite-dimensional general linear group.

A substantial literature has therefore focused on identifying analytic conditions under which a scalar determinant behaves “as expected.” In particular, recent work has shown that under trace-class hypotheses near the identity, the Fredholm determinant trivializes on commutators. This completes the perturbative Fredholm picture.

Here’s the structural observation: the obstruction is not analytic—it’s categorical. In finite dimensions, scalar-valuedness and universality coincide only because the abelianization is one-dimensional. In infinite dimensions, the abelianization is much larger. Forcing it into (\mathbb{C}^\times) necessarily collapses information.

The natural fix is to enlarge the codomain.

I construct a determinant (\Delta) on a global, non-perturbative group of invertible operators characterized by logarithmic traceability. The definition uses an extended trace that combines the normal trace with Dixmier-type contributions, and it lands not in (\mathbb{C}^\times) but in a complex torus. This determinant is multiplicative, invariant under conjugation, functorial with respect to *-homomorphisms, and—crucially—its kernel coincides with the commutator subgroup (up to closure). In particular,
[
\Delta(ABA^{-1}B^{-1}) = 1
]
for all (A,B) in its domain, without trace-class proximity assumptions.

On trace-class perturbations of the identity, (\Delta) reduces to the Fredholm determinant. On finite von Neumann algebras, it agrees with the Fuglede–Kadison determinant. Along smooth paths, it coincides with the de la Harpe–Skandalis determinant. Existing theories appear as boundary cases of a single, functorial framework.

The takeaway is simple: infinite dimensions don’t need a “better” scalar determinant. They need the right target. Once the codomain is allowed to reflect the true size of the abelianization, the commutator identity becomes structural rather than fragile.