Unitary Representations of Groups, Duals, and
                    Characters

Bekka, Bachir; Harpe, Pierre de la

doi:10.1090/surv/250

Cited by 34 publications

(24 citation statements)

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“…In particular, the following non-discrete version of Day's problem is open and intriguing: Is every amenable second countable tdlc group elementary (in the sense of Section 2)? The unitary representation theory also reveals a fundamental dichotomy between locally compact groups of type I (roughly speaking, those for which the problem of classifying the irreducible unitary representations up to equivalence is tractable) and the others (see [10,35,51]). Algebraic characterizations of type I groups are also desirable.…”

Section: Future Directionsmentioning

confidence: 99%

A totally disconnected invitation to locally compact groups

Caprace¹,

Willis²

2021

Preprint

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Section: Future Directionsmentioning

confidence: 99%

A totally disconnected invitation to locally compact groups

Caprace¹,

Willis²

2021

Preprint

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“…We prove here some preliminary results for later use. For the notions and notation in

C^{*}

‐algebras and topology, we refer the reader to [2, 6 10, 30].…”

Section: Preliminariesmentioning

confidence: 99%

“…Proof For every

H \in L (G)

, we denote by

j_{H} : G \to G / H

its corresponding quotient map. We define

\hat{j_{H}} : \hat{G / H} \to \hat{G}, false[σ false] \mapsto false[σ \circ j_{H} false]

and

{(j_{H})}^{*} : Prim false(G / H false) \to Prim false(G false), Ker σ \mapsto Ker false(σ \circ j_{H} false) .

By [2, Proposition 8.C.8], there exists a surjective

*

‐morphism

{(j_{H})}_{*} : C^{*} false(G false) \to C^{*} false(G / H false)

satisfying

σ \circ {(j_{H})}_{*} = σ \circ j_{H}

for every

false[σ false] \in \hat{G / H} ≃ \hat{C^{*} false(G / H false)}

, where

false[σ \circ j_{H} false] \in \hat{G} ≃ \hat{C^{*} false(G false)}

. Therefore

(j_{H}

…”

Section: On Primitively Af‐embeddable C∗‐algebrasmentioning

confidence: 99%

“…Hence, when we extend

π

and

π_{0}

to irreducible

*

‐representations of

C^{*} false(G false)

and

C^{*} false(G / H false)

, respectively, we have

π = π_{0} \circ {(j_{H})}_{*},

where

{(j_{H})}_{*} : C^{*} false(G false) \to C^{*} false(G / H false)

is a surjective

*

‐morphism. (See [2, Proposition 8.C.8].) Since

Prim (G)

T_{1}

, it follows by Lemma 6.4 that

Prim (G / H)

T_{1}

.…”

Section: On Primitively Af‐embeddable C∗‐algebrasmentioning

confidence: 99%

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AF‐embeddability for Lie groups with T1 primitive ideal spaces

Beltiţă

2021

J. London Math. Soc.

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We study simply connected Lie groups G for which the hull-kernel topology of the primitive ideal space Prim(G) of the group C * -algebra C * (G) is T1, that is, the finite subsets of Prim(G) are closed. Thus, we prove that C * (G) is AF-embeddable. To this end, we show that if G is solvable and its action on the centre of [G, G] has at least one imaginary weight, then Prim(G) has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with T1 ideal spaces are strongly quasi-diagonal.

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“…• CCR groups (See [BH19]), such as connected semisimple Lie groups or reductive algebraic groups over local fields. • Nilpotent groups (see [LW95]).…”

Section: Introductionmentioning

confidence: 99%

Evanescent affine isometric actions and weak identity excluding groups

Marrakchi¹

2021

Preprint

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We investiguate a property of affine isometric actions on Hilbert spaces called evanescence. Evanescent actions are the extreme opposite of irreducible actions. Every affine isometric action decomposes naturally into an evanescent part and an irreducible part. We study when this decomposition is unique. We also study when an action that has almost fixed points is automatically evanescent. We relate these questions to the identity excluding property for groups. We also relate them to the finiteness of the von Neumann algebras generated by the linear part of the action.

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Unitary Representations of Groups, Duals, and Characters

Cited by 34 publications

References 0 publications

A totally disconnected invitation to locally compact groups

A totally disconnected invitation to locally compact groups

AF‐embeddability for Lie groups with T1 primitive ideal spaces

Evanescent affine isometric actions and weak identity excluding groups

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