## Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform

### Volume 118 / 2010

#### Abstract

For any connected Lie group $G$ and any Laplacian $\Lambda=X^2_1 +\cdots+X^2_n\in{\mathfrak U}{\mathfrak g}$ ($X_1,\ldots,X_n$ being a basis of ${\mathfrak g}$) one can define the commutant ${\mathfrak B}={\mathfrak B}(\Lambda)$ of $\Lambda$ in the convolution algebra ${\cal L}^1(G)$ as well as the commutant ${\mathfrak C}(\Lambda)$ in the group $C^*$-algebra $C^*(G)$. Both are involutive Banach algebras. We study these algebras in the case of a “distinguished Laplacian” on the “Iwasawa part $AN$” of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group. As a consequence one sees that both algebras are commutative (which is not immediate from the definition), ${\mathfrak B}$ is $C^*$-dense in ${\mathfrak C}$, and ${\mathfrak B}$ is a completely regular symmetric Wiener algebra. As a byproduct of our approach we give another proof of the injectivity of Harish-Chandra's spherical Fourier transform, which is based on a theorem on $C^*$-algebras of solvable Lie groups (due to N. V. Pedersen). The article closes with some open questions for more general solvable Lie groups. To some extent the article is written with a view to these questions, that is, we try to apply, as much as possible (at the moment), methods which work also outside the semisimple context.