Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras

We revisit the old result that biflat Banach algebras have the same cyclic cohomology as $\mathbb C$, and obtain a quantitative variant (which is needed in forthcoming joint work of the author). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [Connes,1985] and [Helemskii,1992].

💡 Research Summary

The paper revisits the classical theorem that biflat Banach algebras have the same cyclic cohomology as the ground field ℂ, but it goes further by providing explicit norm estimates for the homotopy operators that underlie this result. The author’s motivation is two‑fold: first, to obtain quantitative bounds that are required in forthcoming joint work, and second, to present a proof that does not rely on the Connes‑Tsygan exact sequence, even though the construction is heavily inspired by the original ideas of Connes (1985) and the homological framework of Helemskii (1992).

Background and definitions.
A Banach algebra A is called biflat if there exists a continuous A‑bimodule map
(\sigma : A \to A \widehat{\otimes} A)
such that (\sigma) is a right inverse of the multiplication map and its operator norm is bounded by a constant M. This constant M is called the biflatness constant. Biflatness is equivalent to the existence of a bounded linear splitting of the canonical projective resolution of A as an A‑bimodule.

Cohomological complexes.
The Hochschild cochain complex is (C^{n}(A)=\mathcal{B}(A^{\widehat{\otimes} n},\mathbb{C})) with the usual Hochschild coboundary (b). The cyclic subcomplex (C_{\lambda}^{n}(A)) consists of cochains invariant under the cyclic operator (\lambda). The standard Connes‑Tsygan long exact sequence relates Hochschild, cyclic, and periodic cyclic cohomology, and is traditionally used to compute (HC^{\bullet}(A)) for biflat algebras.

Splitting maps and homotopy operators.
The core of the paper is the explicit construction of a family of continuous linear maps
(s^{n}: C^{n}(A) \to C^{n-1}(A)) (for (n\ge 1))
that satisfy the homotopy identity
(b s^{n} + s^{n+1} b = \operatorname{id} - \pi^{n}),
where (\pi^{n}) is the projection onto the cyclic subcomplex. Moreover, the maps commute with the cyclic operator: (\lambda s^{n}=s^{n}\lambda). The construction proceeds by iterating the biflat splitting (\sigma) and inserting it into the tensor factors of a cochain. The resulting formulas are completely explicit and involve only the algebraic operations of multiplication, the bimodule structure, and the bounded map (\sigma).

Norm estimates.
Because each step of the construction uses (\sigma) at most once, the operator norm of (s^{n}) can be bounded in terms of the biflatness constant M. The paper proves the quantitative bounds
\