— The paper deals with the reduced semigroup C ∗ -algebra C ∗ r ( Z (cid:2) ϕ Z × ) for the semidirect product Z (cid:2) ϕ Z × of the additive group Z of all integers and the multiplicative semigroup Z × of integers without zero. The semigroup homomorphism ϕ acts from Z × to the endomorphism semigroup of Z as follows. It takes every positive integer to the identity endomorphism of Z and every negative integer to the inversion of Z . The purpose of the paper is to demonstrate that the C ∗ -algebra C ∗ r ( Z (cid:2) ϕ Z × ) and the in ﬁ nite dihedral group D ∞ are closely related. To this end, we prove three results. Firstly, we show that the C ∗ -algebra C ∗ r ( Z (cid:2) ϕ Z × ) is topologically graded over the group D ∞ . In order to obtain this result, we use a general method for constructing topological gradings. Previously, this method was proposed in the study of extensions of semigroups and the reduced semigroup C ∗ -algebras associated with such extensions. As a consequence, the C ∗ - algebra C ∗ r ( Z (cid:2) ϕ Z × ) has a countable family of Fourier coe ﬃ cients which is indexed by the elements of D ∞ . Secondly, we construct a covariant representation of a non-commutative dynamical system de ﬁ ned by means of the group D ∞ into the C ∗ -algebra C ∗ r ( Z (cid:2) ϕ Z × ) . Thirdly, we establish an isomorphism between the C ∗ -algebra C ∗ r ( Z (cid:2) ϕ Z × ) and a crossed product of its C ∗ -subalgebra by the group D ∞ .

A Semigroup C ∗ C ∗ C ∗ -Algebra Related To the In ﬁ nite Dihedral Group

https://link.springer.com/content/pdf/10.1134/S1995080221100152.pdf

A Semigroup $$\boldsymbol{C}^{\mathbf{*}}$$-Algebra Which Is a Free Banach Module

The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra

E V Lipacheva

Papers

Topological grading of semigroup c*-algebras