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We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of 
$\mathbf{U}_{q}$-tilting modules (for any field 
$\mathbb{K}$ and any parameter 
$q\in \mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types 
$\mathbf{A}$ and 
$\mathbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.
$\ast $-SEMISIMPLICITY OF THE 
${\ell }^{1} $-ALGEBRA ON AN ABELIAN 
$\ast $-SEMIGROUP
Towards an involutive analogue of a result on the semisimplicity of 
${\ell }^{1} (S)$ by Hewitt and Zuckerman, we show that, given an abelian 
$\ast $-semigroup 
$S$, the commutative convolution Banach 
$\ast $-algebra 
${\ell }^{1} (S)$ is 
$\ast $-semisimple if and only if Hermitian bounded semicharacters on 
$S$ separate the points of 
$S$; and we search for an intrinsic separation property on 
$S$ equivalent to 
$\ast $-semisimplicity. Very many natural involutive analogues of Hewitt and Zuckerman’s separation property are shown not to work, thereby exhibiting intricacies involved in analysis on 
$S$.
