arXiv subjects

Yemon Choi

Publications and source records attributed to Yemon Choi.

39 records · Page 3Linked to original sources

arXiv2007

Injective convolution operators on ${\ell}^{\infty}(Γ)$ are surjective

Let $Γ$ be a discrete group and let $f \in \ell^1(Γ)$. We observe that if the natural convolution operator $ρ_f:\ell^{\infty}(Γ)\to \ell^{\inf ty}(Γ)$ is injective, then f is invertible in $\ell^1(Γ)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra $\ell^1(Γ)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^p$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $Γ$, and give some partial results.

math.FA↗

arXiv2007

Biflatness of ${\ell}^1$-semilattice algebras

Building on an old result of Duncan and Namioka, we show that the ${\ell}^1$-convolution algebra of a semilattice $S$ is biflat precisely when $S$ is uniformly locally finite. The proof shows in passing that for such $S$ the convolution algebra is isomorphic to ${\ell}^1(S)$ with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.

math.FA↗

arXiv2006

Simplicial homology and Hochschild cohomology of Banach semilattice algebras

The ${\ell}^1$-convolution algebra of a semilattice is known to have trivial cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is symmetric. We ex tend this result to all cohomology groups of degree $\geq 1$ with symmetric coef ficients. Our techniques prove a stronger splitting result, namely that the spli tting can be made natural with respect to the underlying semilattice.

math.FA↗