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- Title
Extensions of semigroups by the dihedral groups and semigroup C∗-algebras.
- Authors
Lipacheva, E. V.
- Abstract
We consider the semidirect product Z ⋊ φ Z × of the additive group Z of all integers and the multiplicative semigroup Z × of integers without zero relative to a semigroup homomorphism φ from Z × to the endomorphism semigroup of Z. It is shown that this semidirect product is a normal extension of the semigroup Z × N by the dihedral group, where N is the multiplicative semigroup of all natural numbers. Further, we study the structure of C ∗ -algebras associated with this extension. In particular, we prove that the reduced semigroup C ∗ -algebra of the semigroup Z ⋊ φ Z × is topologically graded over the dihedral group. As a consequence, there exists a structure of a free Banach module over the reduced semigroup C ∗ -algebra of Z × N in the underlying Banach space of the reduced semigroup C ∗ -algebra of Z ⋊ φ Z × .
- Subjects
C*-algebras; ENDOMORPHISMS; NATURAL numbers; BANACH spaces; HOMOMORPHISMS; INTEGERS
- Publication
Journal of Algebra & Its Applications, 2024, Vol 23, Issue 2, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498824500221