The $$C^*$$-Algebra of the Heisenberg Motion Groups $${{\mathbb {T}}}^n\ltimes {\mathbb {H}}_n$$

The C*-algebra of Boidol's group

“…The method of describing group C*-algebras as algebras of operator fields defined on the dual spaces was first introduced in [4] and [10]. In serial, the C*-algebra of ax + b-like groups [11], the C*-algebras of the Heisenberg groups and of the threadlike groups [13], the C*-algebras of the affine automorphism groups G n,µ in [7], [8], and the C*-algebra of the group T ⋉ H 1 [14] were all characterised as algebras of operator fields defined on the corresponding spectrum of the groups. In this way, the C*-algebra of every exponential Lie group of dimension less than or equal to 4 has been explicitly determined with one exception, namely Boidol's group G, which is an extension of the Heisenberg group by the reals with the roots 1 and −1.…”