On Extensions of Semigroups and Their Applications to Toeplitz Algebras

The paper deals with the normal extensions of cancellative commutative semigroups andthe Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup $S$ one means the reduced semigroup $C^*$-algebra $C^*_r(S)$. We study the normal extensions of cancellative commutative semigroups by the additive group $\mathbb{Z}_n$ of integers modulo $n$. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup $S$ and its normal extension $L$ generated by one element, there exists a natural embedding the semigroup $C^*$-algebra $C^*_r(S)$ into $C^*_r(L)$. The semigroup $C^*$-algebra $C^*_r(L)$ is topologically $\mathbb{Z}_n$-graded.The results in the paper are announced without proofs