Last-slot normal cocycles on a tracial algebra without abelian corners

Throughout, $M$ is a unital C*-algebra with a partial order making it star-ordered, $X$ is a complex Banach space, and $e:X^*\to M$ is an isometric conjugate-linear isomorphism, so $M$ is a von Neumann algebra with predual $X$ (the Lean hypothesis e). A family $(a_i)$ converges in the predual sense to $b$ (PredualTendsto) when $e^{-1}(a_i)(z)\to e^{-1}(b)(z)$ for every $z\in X$. Here $\tau$ is a positive linear functional on $M$ with $\tau(1)=1$ that lies in the predual (TraceInPredual: $\tau(x)=\overline{e^{-1}(x)(z)}$ for some $z\in X$), is tracial, and is faithful ($\tau(a^*a)=0$ implies $a=0$), and $\|x\|_2=\tau(x^*x)^{1/2}$ is the norm of τ.toPreGNS x. The hypothesis NoAbelianCorners M says that for every nonzero projection $p$ the corner algebra $\{a:pa=a=ap\}$ contains two noncommuting elements. A bounded operator $L$ on $M$ has a preadjoint (HasPreadjoint e L) when for each $z\in X$ there is $z'\in X$ with $e^{-1}(L(a))(z)=e^{-1}(a)(z')$ for all $a$, i.e. when it is ultraweakly continuous. Assume NoAbelianCorners M, and let $f$ be a bounded cocycle of degree $n+1$ on $M$ such that $x\mapsto f(a,x)$ has a preadjoint for each $a\in M^n$. Then there is $g$ of degree $n$ with $\|g\|\le\|f\|$ and $\partial g=f$. No separability is assumed. Here a cochain of degree $k$ is a bounded complex $k$-linear map $M^k\to M$, and $\partial$ (differentialValue) is the Hochschild differential $(\partial f)(v_0,\dots,v_k)=v_0f(v_1,\dots,v_k)+\sum_{j<k}(-1)^{j+1}f(\dots,v_jv_{j+1},\dots)+(-1)^{k+1}f(v_0,\dots,v_{k-1})v_k$.

Formalization Note The statement is verbatim from OpenAI's Lean development, with the listed Definitions imported in place of the upstream modules; the definitions are supporting code for this statement. Only the keyword lemma, where used upstream, becomes theorem, and a leading include ... in or omit ... in moves to the preamble. It is a step of the Lean proof of the vanishing of higher bounded Hochschild cohomology.