Abstract

The note is concerned with inductive sequences of Toeplitz algebras. The Toeplitz algebra is the \(C^{*}\)-subalgebra in the algebra of all bounded linear operators. This subalgebra is generated by the right shift operator on the Hilbert space of all square summable complex-valued functions defined on the additive semigroup of non-negative integers. We study the inductive sequences of Toeplitz algebras whose bonding \(\ast\)-homomorphisms are defined by arbitrary sequences of natural numbers. The inductive limits of such sequences are the reduced semigroup \(C^{*}\)-algebras generated by representations for semigroups of non-negative rational numbers. We consider the limit \(\ast\)-endomorphisms of these inductive limits. Such an endomorphism is induced by a morphism between two copies of the same inductive sequence of Toeplitz algebras. We give the necessary and sufficient conditions for these endomorphisms to be \(\ast\)-automorphisms of \(C^{*}\)-algebras. These criteria are formulated in algebraic, number-theoretical and functional terms.

中文翻译：

---

Toeplitz代数的归纳序列和极限自同构

摘要

该注释与托普利兹代数的归纳序列有关。Toeplitz代数是所有有界线性算子的代数中的\（C ^ {*} \）-子代数。该子代数是由在非负整数的加法半群上定义的所有平方可加复数值函数的希尔伯特空间的右移运算符生成的。我们研究了Toeplitz代数的归纳序列，它们的键\（\ ast \）-同态由自然数的任意序列定义。这样的序列的归纳极限是由非负有理数的半群的表示形式生成的简化半群\（C ^ {*} \）-代数。我们考虑极限\（\ ast \）这些归纳极限的内同态。这种内同形是由Toeplitz代数的相同归纳序列的两个副本之间的同构引起的。我们给出这些内同构为\（\ ast \） - \（C ^ {*} \）-代数的自同构的必要和充分条件。这些标准用代数，数论和功能术语表述。