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In this paper, we describe the  C *‐algebras associated to the Baumslag–Solitar groups with the ordering defined by the usual presentations. These are Morita equivalent to the crossed product  C *‐algebras obtained by letting the group act on its  directed boundary . We use the method of categories of paths to define the algebras, and to deduce the presentation by generators and relations. We obtain a complete description of the Toeplitz algebras, and we compute the  K ‐theory of the Cuntz–Kreiger algebras.

C *‐algebras for categories of paths associated to the Baumslag–Solitar groups