  The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup
${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of
$\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution
algebra associated to $e_{\mathcal{P}}$ is called a Peter-Weyl Iwahori algebra.
We show any Peter-Weyl Iwahori algebra is Morita equivalent to the
Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori
algbera have a natural $\mathbb{C}^\star$-algebra structure, and the Morita
equivalence preserves irreducible hermitian and unitary modules. Both algebras
have another anti-involution denoted as $\bullet$, and the Morita equivalence
preserves irreducible and unitary modules for the $\bullet$-involution.
