Algebras of conjugacy classes of partial elements

In 2001 Ivanov and Kerov associated with the infinite permutation group $S_\infty$ certain commutative associative algebra $A_\infty$ called the algebra of conjugacy classes of partial elements. A standard basis of $A_\infty$ is labeled by Yang diagrams of all orders. Mironov, Morozov, Natanzon, 2012, have proved that the completion of $A_\infty$ is isomorphic to the direct product of centers of group algebras of groups $S_n$. This isomorphism was explored in a construction of infinite dimensional Cardy-Frobenius algebra corresponding to asymptotic Hurwitz numbers. In this work algebras of conjugacy classes of partial elements are defined for a wider class of infinite groups. It is proven that completion of any such algebra is isomorphic to the direct product of centers of group algebras of relevant subgroups.