The paper deals with the normal extensions of cancellative commutative semigroups and the 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 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 ℤ_n-graded. The results in the paper are announced without proofs.

中文翻译：

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半群的扩展及其在Toeplitz代数中的应用

本文讨论了可交换交换半群的正则扩展以及这些半群的Toeplitz代数。对于半群S的Toeplitz代数，表示简化的半群C *-代数C ^* _r（S）。我们通过加群ℤ研究消可交换半群的正规扩张_ñ整数模ñ。而且，我们假设这样的扩展是由一个元素生成的。我们提出了一种一般的方法来构造半群的正态扩展，该半群不包含非平凡的子群。给定半群的Grothendieck群和所有整数的群都参与此构造。给出了非负整数的加法半群的此类扩展的示例。用数论术语给出了一个元素所生成的正态扩展与数字半群同构的标准。关于托普利兹代数的结果如下。对于一个可交换交换半群S及其由一个元素生成的法向扩展L，存在自然嵌入半群C *-代数C^* _r（ S）转换为C ^* _r（ L）。半群C * -代数Ç ^* _{- [R} （大号）是拓扑ℤ _Ñ -graded。本文中的结果未经证实就宣布。