Alphabeta MathHow statement and proof provenance work
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Continuous positive-type functions and normalization
Definition
Let be a topological group with identity . A continuous function is of positive type if, for every integer , every list (repetitions allowed), and every list (zero values allowed), the matrix is positive semidefinite. Write for the set of continuous functions of positive type and for the normalized positive-type functions. For , the notation means that both and belong to .
Depends on
Used by
- Normalized positive type and pointed cyclic unitary representations Corollary
- A bounded continuous normalized function that is not of positive type Counterexample
- GNS representation of a continuous unitary character Example
- Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example
- The positive-type Gaussian on the real line and its cyclic model Example
- Diagonal unitary coefficients have positive type Lemma
- Dominated positive type and positive commutant contractions Lemma
- Nonscalar commutant contractions and convex decompositions Lemma
- Positive-type functions define the GNS pre-Hilbert form Lemma
- The GNS translation action is unitary and strongly continuous Lemma
- Extreme normalized positive type is equivalent to irreducible GNS Theorem
Dependency tree · two levels
4 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.4.1 and Proposition C.4.2, Appendix C, printed pp. 373–374 (standard reference, not scraped)
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Definition 1.B.1, Chapter 1 §1.B, printed p. 26 (standard reference, not scraped)