标题： 二次子积系统，自由积及其C*-代数 显示英文标题

标题： Quadratic subproduct systems, free products, and their C*-algebras

摘要： 受二次代数、非交换几何和算子理论之间相互作用的启发，我们引入了Hilbert空间的二次子乘积系统的概念。 具体来说，我们研究由有限数量的非交换变量中的复二次多项式诱导的子乘积系统，并描述它们的Toeplitz代数和Cuntz-Pimsner代数。 受到分级结合代数理论的启发，我们在子乘积系统的范畴内定义了一个自由积运算，并证明这对应于Toeplitz代数的约化自由积。 最后，我们得到了一类大型二次子乘积系统的Toeplitz代数和Cuntz-Pimsner代数的K-理论的结果。 显示英文摘要

摘要： Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz--Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz--Pimsner algebras of a large class of quadratic subproduct systems.