Weak Containment and Induced Representations of Groups

1964 ◽  
Vol 110 (3) ◽  
pp. 424 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Induced Representations ◽  
Representations Of Groups ◽  
Weak Containment

scholarly journals Induction for locally compact quantum groups revisited

2019 ◽  
Vol 150 (2) ◽  
pp. 1071-1093
Author(s):  
Mehrdad Kalantar ◽  
Paweł Kasprzak ◽  
Adam Skalski ◽  
Piotr M. Sołtan
Keyword(s):  
Quantum Groups ◽  
General Setting ◽  
Locally Compact ◽  
Induced Representations ◽  
Open Quantum ◽  
Locally Compact Quantum Groups ◽  
Weak Containment ◽  
Compact Quantum Groups

AbstractIn this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

scholarly journals Weak Containment by Restrictions of Induced Representations

2018 ◽  
Vol 2020 (7) ◽  
pp. 2034-2053
Author(s):  
Matthew Wiersma
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Discrete Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Locally Compact ◽  
Induced Representations ◽  
C Algebras ◽  
Weak Containment

Abstract A QSIN group is a locally compact group G whose group algebra $\mathrm L^{1}(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if G is a QSIN group, H is a closed subgroup of G, and $\pi \!: H\to \mathcal B(\mathcal{H})$ is a unitary representation of H, then $\pi$ is weakly contained in $\Big (\mathrm{Ind}_{H}^{G}\pi \Big )|_{H}$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if G is a QSIN group which contains a copy of $\mathbb{F}_{2}$ as a closed subgroup, then $\mathrm C^{\ast }(G)$ is not locally reflexive and $\mathrm C^{\ast }_{r}(G)$ does not admit the local lifting property. Further applications are drawn to the “(weak) extendability” of Fourier spaces $\mathrm A_{\pi }$ and Fourier–Stieltjes spaces $\mathrm B_{\pi }$.

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

Narratives shape cognitive representations of immigrants and policy preferences

2018 ◽  
Author(s):  
Joel Eduardo Martinez ◽  
Lauren Feldman ◽  
Mallory Feldman ◽  
Mina Cikara
Keyword(s):  
Immigration Policy ◽  
Mexican Immigrants ◽  
Policy Preferences ◽  
Cognitive Representations ◽  
Induced Representations ◽  
The Social

Scholars from across the social and media sciences have issued a clarion call to address a recent resurgence in criminalized characterizations of immigrants. Do these characterizations meaningfully impact individuals’ beliefs about immigrants and immigration? Across two online convenience samples (N = 1,054 adult U.S. residents), we applied a novel analytic technique to test how different narratives—criminal, achievement, struggle-oriented—impact cognitive representations of German, Russian, Syrian, and Mexican immigrants and the concept of “immigrants” in general. All stories featured male targets. Achievement stories homogenized individual immigrant representations whereas both criminal and struggle-oriented stories racialized them along a white/non-white axis: Germany clustered with Russia, Syria with Mexico. However, criminal stories were unique in making our most egalitarian participants’ representations as differentiated as our least egalitarian participants’. Narratives about individual immigrants also generalized to update representations of nationality groups. Most important, narrative-induced representations correlated with immigration policy preferences: achievement narratives and corresponding homogenized representations promoted preferences for less restriction, criminal narratives for more.

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