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Quadratic Subproduct Systems, Free Products, and Their C*-Algebras
1Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, the Netherlands.
Integral Equations and Operator Theory
|May 4, 2026
Summary
We introduce quadratic subproduct systems and their associated Toeplitz and Cuntz-Pimsner algebras. These systems, derived from quadratic polynomials, offer insights into noncommutative geometry and operator theory.
Area of Science:
- Noncommutative Geometry
- Operator Theory
- Algebraic Structures
Background:
- The study is motivated by the connections between quadratic algebras, noncommutative geometry, and operator theory.
- Existing research highlights the importance of subproduct systems in functional analysis.
Purpose of the Study:
- To introduce and investigate quadratic subproduct systems of Hilbert spaces.
- To describe the Toeplitz and Cuntz-Pimsner algebras associated with these systems.
- To explore the K-theory of these algebras.
Main Methods:
- Definition of quadratic subproduct systems using complex quadratic polynomials in noncommuting variables.
- Description of the induced Toeplitz and Cuntz-Pimsner algebras.
- Introduction of a free product operation for subproduct systems.
- Analysis of the K-theory for a broad class of these systems.
Main Results:
- The Toeplitz and Cuntz-Pimsner algebras for quadratic subproduct systems are characterized.
- A free product operation on subproduct systems is defined and shown to correspond to the reduced free product of their Toeplitz algebras.
- New results on the K-theory of these algebras are established.
Conclusions:
- Quadratic subproduct systems provide a novel framework linking algebraic and analytic structures.
- The study advances the understanding of noncommutative structures and their applications in operator theory.
- The K-theoretic results offer valuable tools for further research in related fields.
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