On Extensions of Semigroups and Their Applications to Toeplitz Algebras

被引：0
作者：
S. A. Grigoryan
R. N. Gumerov
E. V. Lipacheva
机构：
[1] Kazan Power Engineering University,Chair of Higher Mathematics
[2] Kazan (Volga Region) Federal University,Chair of Mathematical Analysis, Lobachevskii Institute of Mathematics and Mechanics
来源：
Lobachevskii Journal of Mathematics | 2019年 / 40卷
关键词：
cancellative semigroup; *-algebraic bundle; *-grading; extension of semigroup; exact sequence; Fell bundle; normal extension generated by element; graded ; -algebra; Grothendieck group; numerical semigroup; reduced semigroup ; -algebra; semi-group of non-negative integers; Toeplitz algebra; topologically graded ; *-;
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中图分类号：
学科分类号：
摘要：
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 Cr*(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤn 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 Cr*(S) into Cr*(L). The semigroup C*-algebra Cr*(L) is topologically ℤn-graded. The results in the paper are announced without proofs.
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页码：2052 / 2061
页数：9
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