Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Cyclic vector and cyclic unitary representation
Definition
Let be a unitary representation. A vector is cyclic if The representation is cyclic if it has a cyclic vector. The representation on the zero Hilbert space is cyclic, with its unique vector as a cyclic vector. In particular this convention permits the zero representation in the unnormalized GNS statement; irreducibility remains defined only for nonzero Hilbert spaces.
Depends on
Used by
- Normalized positive type and pointed cyclic unitary representations Corollary
- GNS representation of a continuous unitary character Example
- The positive-type Gaussian on the real line and its cyclic model Example
- Nonscalar commutant contractions and convex decompositions Lemma
- GNS construction for a continuous positive-type function Theorem
- Uniqueness of the pointed cyclic GNS representation Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition C.4.8, Appendix C, printed p. 375 (standard reference, not scraped)
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Definition 1.B.4, §1.B (standard reference, not scraped)