Secondary invariants for Frechet algebras and quasihomomorphisms

A Frechet algebra endowed with a multiplicatively convex topology has two types of invariants: homotopy invariants (topological K-theory and periodic cyclic homology) and secondary invariants (multiplicative K-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann-Roch-Grothendieck theorem relating direct images for homotopy and secondary invariants of Frechet m-algebras under finitely summable quasihomomorphisms.

💡 Research Summary

The paper investigates two layers of invariants associated with Fréchet algebras equipped with a multiplicatively convex topology, often called Fréchet m‑algebras. The first layer consists of homotopy invariants – topological K‑theory (KU) and periodic cyclic homology (HP) – which remain unchanged under continuous deformations. The second layer, called secondary or “2‑ary” invariants, includes multiplicative K‑theory (KM) and non‑periodic cyclic homology (HC). These capture finer, degree‑dependent information that is lost in the periodic setting.

The central technical device introduced is a finite‑summable quasihomomorphism. Given two Fréchet m‑algebras A and B, a linear map φ: A → B̂ ⊗ 𝓚 (with 𝓚 the algebra of compact operators) is called an n‑summable quasihomomorphism if for every a∈A and b∈B̂ ⊗ 𝓚 the commutator φ(a)·b − b·φ(a) belongs to the Schatten class L^{n+1}. This condition generalizes Kasparov’s KK‑theory bimodules to the Fréchet context and provides a controlled way to measure how far φ is from being a genuine homomorphism.

Using φ, the author constructs push‑forward (direct image) maps on both layers of invariants. On K‑theory, φ induces a map
 f_* : K_(A) → K_{+n}(B)
which shifts degree by the summability order n. The construction relies on the operator‑valued trace associated with the Schatten class condition and on a careful analysis of the induced Kasparov‑type module.

On the secondary side, the paper defines a non‑periodic Chern character ch: K_(A) → HC_(A) and shows that φ gives rise to a compatible map
 f_! : HC_(A) → HC_{+n}(B).
The map f_! is built from a trace functional τ_φ that evaluates the φ‑twisted commutators and respects the Connes differential b+B.

The main theorem is a Riemann‑Roch‑Grothendieck (RRG) formula for Fréchet m‑algebras:
 ch_B ∘ f_* = f_! ∘ ch_A.
In words, the Chern character intertwines the homotopy‑level push‑forward with the secondary push‑forward. The proof uses filtered chain complexes, continuity estimates for the boundary operators, and precise Schatten‑class bounds to show that the correction terms vanish.

A further contribution is the extension of the theory to multiplicative K‑theory. The author defines a multiplicative Chern character ch^{np}: KM_(A) → HC_{+2ℤ+1}(A) and proves that the same RRG compatibility holds for the corresponding push‑forwards. This demonstrates that degree‑shifted K‑theory and non‑periodic cyclic homology form a coherent pair under finite‑summable quasihomomorphisms.

Finally, the paper discusses applications. The results provide a framework for degree‑dependent index theorems in non‑commutative geometry, allowing one to compute indices that retain information about the summability order. They also suggest new tools for studying spectral invariants such as beta functions or logarithmic terms that appear in quantum field theory, where the underlying algebras are often Fréchet‑type. In summary, the work establishes a robust bridge between homotopy invariants and secondary invariants for Fréchet m‑algebras, extending the classical RRG paradigm to a non‑commutative, analytically delicate setting.