During this publication the authors boost a concept of unfastened noncommutative services, in either algebraic and analytic settings. Such features are outlined as mappings from sq. matrices of all sizes over a module (in specific, a vector area) to sq. matrices over one other module, which appreciate the scale, direct sums, and similarities of matrices. Examples comprise, yet should not restricted to, noncommutative polynomials, strength sequence, and rational expressions. Motivation and concept for utilizing the idea of loose noncommutative capabilities usually comes from loose chance. a big program quarter is "dimensionless" matrix inequalities; those come up, e.g., in numerous optimization difficulties of process engineering. between different comparable components are these of polynomial identities in earrings, formal languages and finite automata, quasideterminants, noncommutative symmetric features, operator areas and operator algebras, and quantum keep an eye on.
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