Representations of the Banach algebra \(\ell ^ 1(S)\)
Let S be a monoid equipped with an involution *. Then the Banach space \(\ell^ 1(S)\) becomes a Banach *-algebra when multiplication is defined by convolution. This algebra is *-semisimple if and only if the bounded *-representation of S, on Hilbert spaces, separate points of S. We give a new proof of this result using properties of von Neumann algebras. It is also shown that when S is the free monoid on a countable set of generators then \(\ell^ 1(S)\) has a separating sequence of finite dimensional irreducible *-representations.
free monoid on a countable set of generators
separating sequence of finite dimensional irreducible *-representations
scientific article; zbMATH DE number 4030264