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Strongly continuous unitary representations, invariant linear subspaces and intertwiners

Definition

Let $G$ be a topological group and $H$ a complex Hilbert space. A unitary representation of $G$ on $H$ is a group homomorphism $π : G \to U (H)$ , where $U (H)$ is the group of bijective complex-linear isometries of $H$ , such that the orbit map $g \mapsto π (g) v$ is norm-continuous for every $v \in H$ . This condition is strong continuity. A closed linear subspace $M \subseteq H$ is invariant if $π (g) M = M$ for every $g \in G$ . The representation is irreducible if $H \neq {0}$ and its only closed invariant linear subspaces are ${0}$ and $H$ .

For unitary representations $π$ on $H$ and $ρ$ on $K$ , a bounded intertwiner is a bounded linear operator $T : H \to K$ satisfying $T π (g) = ρ (g) T (g \in G) .$ The representations are unitarily equivalent if there is a unitary intertwiner between them. The commutant of $π$ is $π (G)^{'} = {T \in B (H) : T π (g) = π (g) T for every g \in G} .$

Depends on

Used by

Mackey little-group reduction for an abelian normal subgroup Corollary
Normalized positive type and pointed cyclic unitary representations Corollary
A nontransitive system with two orbits is not classified by one stabilizer Counterexample
The integers do not have property (T) Counterexample
The regular representation of R is not a Hilbert direct sum of irreducibles Counterexample
The unitary dual need not be Hausdorff Counterexample
Almost invariant vectors for a unitary representation Definition
Compact-group isotypic projection Definition
Connected Poincaré covers and positive translation energy Definition
Continuous covariant model and measurable completion Definition
Cyclic vector and cyclic unitary representation Definition
Direct integrals of unitary representations Definition
Factor (primary) representations Definition
Haar averaging of bounded operators as a weak operator integral Definition
Hilbert direct sums of unitary representations Definition
Kazhdan pairs, Kazhdan sets and Kazhdan constants Definition
Kazhdan's property (T) Definition
Mackey Borel structure and countable separation of the unitary dual Definition
Matrix coefficient of a unitary representation Definition
Regular Weyl systems and irreducibility Definition
Relative property (T) for a pair and relative Kazhdan pairs Definition
Representative functions on a compact group Definition
Smooth and K-finite vectors for SL2(R), and the (g,K)-module Definition
Spectral gap for a unitary representation Definition
Systems of imprimitivity for a Borel G-space Definition
Tempered unitary representations Definition
The integrated form of a unitary representation Definition
The unitary dual of a compact group Definition
The unitary dual of a locally compact group Definition
Type I factor representations and type I groups Definition
Unitary equivalence of systems of imprimitivity and of the induced representations Definition
Weak containment of unitary representations Definition
A Kazhdan pair for a compact group via Haar averaging Example
Fell convergence of the characters of the real line Example
GNS representation of a continuous unitary character Example
Isotypic Haar projections specialize to finite character sums Example
Little groups for the real ax+b group and its orientation-preserving subgroup Example
Peter-Weyl for a profinite group Example
Peter-Weyl for an infinite product of finite groups Example
Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example

…and 69 more results.

Dependency tree · two levels

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Sources

Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1 and Propositions A.1.2–A.1.4, Appendix A, printed pp. 305–307 (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, Definition 3.4.1, §3.4, printed p. 106 (standard reference, not scraped)