Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws
definitiondef:bounded-tuple-law-tracial-2026aAnalysisProbabilityA self-adjoint d-tuple in a tracial W*-probability space is a d-tuple of self-adjoint elements of its algebra; its law is the functional on noncommutative polynomials given by the vacuum expectation of their values.Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings
theoremthm:inductive-limit-tracial-w-star-2026aAnalysisA sequence of tracial W*-probability spaces linked by trace-preserving embeddings has an inductive limit: a tracial W*-probability space receiving compatible trace-preserving embeddings of every term, whose images are dense and generate it as a double commutant.Trace-Preserving Embeddings Preserve Operator Norms and Compose
lemmalem:trace-preserving-embedding-basic-2026aAnalysisA trace-preserving embedding of tracial W*-probability spaces preserves operator norms, and the composite of two such embeddings is again one, implemented by the composite isometry.The Conditional Expectation of a Trace-Preserving Embedding: Values in the Subalgebra, Bimodule Property, Trace, Positivity, Contraction and the Jones Projection
lemmalem:conditional-expectation-tracial-2026aAnalysisThe conditional expectation of a trace-preserving embedding takes values in the smaller algebra, is a positive, contractive, trace-preserving bimodule map that is a left inverse of the embedding, and acts on vectors through the orthogonal projection onto the closure of the embedd…Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations
definitiondef:trace-preserving-embedding-2026aAnalysisA trace-preserving embedding is a trace-preserving unital -homomorphism between tracial W-probability spaces; its implementing isometry V gives the conditional expectation b maps to V* b V.A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry
lemmalem:trace-preserving-homomorphism-isometry-2026aAnalysisA trace-preserving unital -homomorphism between tracial W-probability spaces is injective and is implemented by a unique isometry between the Hilbert spaces that maps the cyclic vector to the cyclic vector and intertwines the homomorphism and the conjugations.The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications
lemmalem:nc-law-tracial-w-star-2026aAnalysisFor a noncommutative law, the left multiplication operators with the vacuum form a cyclic tracial operator algebra whose conjugation is the GNS conjugation and whose W*-closure is the tracial algebra of the law; the variables realise the law in the vacuum state.Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors
theoremthm:tracial-w-star-standard-form-2026aAnalysisIn a tracial W*-probability space the commutant is the conjugate of the algebra, right-bounded vectors come from elements of the algebra, the trace is faithful, and the conjugation-fixed vectors form a real Hilbert space in which the vectors of self-adjoint elements are dense.The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant
theoremthm:tracial-w-star-closure-2026aAnalysisThe commutant of the right action of a cyclic tracial operator algebra contains the algebra, is a tracial W*-probability space for the same vector, equals the double commutant of the algebra, has commutant equal to its own conjugate, and contains an element with any prescribed ri…The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra
lemmalem:cyclic-tracial-conjugation-2026aAnalysisThe trace of a cyclic tracial operator algebra is a faithful tracial state, the cyclic vector separates the algebra and its commutant, the map sending S Omega to S* Omega extends to a conjugation, and conjugating the algebra by it gives right multiplications, which lie in the com…- A tracial W*-probability space is a cyclic tracial operator algebra whose algebra equals its double commutant.
Cyclic Tracial Operator Algebras and Their Traces
definitiondef:cyclic-tracial-operator-algebra-2026aAnalysisA cyclic tracial operator algebra is a unital *-algebra of bounded operators on a complex Hilbert space together with a cyclic unit vector whose vector state is tracial on the algebra; that vector state is its trace.Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation
lemmalem:commutant-basic-2026aAnalysisCommutants are unital algebras closed under weak operator limits and under adjoints when the set is; taking commutants reverses inclusions, a set lies in its double commutant and the triple commutant equals the commutant; conjugation by a conjugation commutes with taking commutan…The Commutant of a Set of Bounded Operators on a Complex Hilbert Space
definitiondef:commutant-complex-hilbert-2026aAnalysisThe commutant of a set of bounded operators consists of the bounded operators commuting with each of them; the double and triple commutants are iterated commutants.- A conjugation of a complex Hilbert space is an additive, conjugate-homogeneous involution that reverses inner products; conjugating an operator by it gives the map J A J.
Bounded Linear and Conjugate-Linear Maps on a Dense Subspace of a Complex Hilbert Space Extend Uniquely
lemmalem:dense-subspace-extension-complex-hilbert-2026aAnalysisA bounded linear or conjugate-linear map on a dense linear subspace of a complex Hilbert space extends uniquely to a continuous map on the whole space with the same bound, isometries extending to isometries; bounded operators agreeing weakly on a dense subspace are equal.- Standing notation for L2 noncommutative laws: the completion and its metric, affine data and push-forwards, moments, couplings and cost, with the basic results in force.
The Distance between Square-Integrable Noncommutative Laws is the Infimum of the Cost over Their Couplings
theoremthm:nc-l2-wasserstein-couplings-2026aAnalysisAny two L2 laws have L2 couplings, and the squared completion distance between them equals the infimum of the cost over those couplings; almost optimal couplings are obtained by chain gluing of optimal couplings of bounded approximations.Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling
lemmalem:nc-l2-laws-calculus-2026aAnalysisOn L2 laws, push-forwards and moments agree with the bounded ones, satisfy the same Lipschitz estimates, compose functorially and obey the same moment formulas and positivity; every joint law couples its marginals with cost at least the squared distance, and the diagonal coupling…Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost
definitiondef:nc-l2-laws-2026aAnalysisThe L2 laws of d variables form the metric completion of the noncommutative laws under the Wasserstein distance; affine push-forwards and first and quadratic moments are defined on it by continuous extension, and couplings and their cost through the marginal and difference data.