Quantum Wreath Products of Transformation Semigroups: Associativity, Distributivity, and Faithful Embeddings into C*-Algebras

Abstract

Connecting discrete semigroup theory with quantum C*-algebras requires a rigorous account of how semigroup constructions are preserved in operator-algebraic settings. This study develops a framework for embedding the direct and wreath products of transformation semigroups into quantum C*-algebras. Through algebraic verification and operator-theoretic construction, the study first establishes that the wreath product of transformation semigroups is associative and distributes over the direct product under appropriate conditions. Faithful representations of these products are then constructed as operators on Hilbert spaces, yielding explicit C*-algebraic isomorphisms. The results show that the direct product admits an embedding as a direct sum of C*-algebras, whereas the wreath product corresponds to a crossed product involving tensor products and the C*-algebras of the acting semigroup. They further demonstrate that semigroup-level isomorphisms lift to the associated C*-algebras and that the wreath product distributes over direct sums in the C*-algebraic setting. Necessary and sufficient conditions are established under which these embeddings preserve both algebraic structures and operator-theoretic properties. By connecting transformation semigroup constructions with noncommutative geometry, this framework provides an operator-algebraic foundation for analyzing direct and wreath products through faithful quantum representations.