C*-norms for tensor products of discrete group C*-algebras

Let $\Gamma$ be a discrete group. We show that if $\Gamma$ is nonamenable, then the algebraic tensor products $C^*_r(\Gamma)\otimes C^*_r(\Gamma)$ and $C^*(\Gamma)\otimes C^*_r(\Gamma)$ do not admit unique $C^*$-norms. Moreover, when $\Gamma_1$ and $\Gamma_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(\Gamma_1)\otimes C^*_r(\Gamma_2)$ and $C^*(\Gamma_1)\otimes C_r^*(\Gamma_2)$ admit $2^{\aleph_0}$ $C^*$-norms. Analogues of these results continue to hold when these familiar group $C^*$-algebras are replaced by appropriate intermediate group $C^*$-algebras.Comment: 6 pages, v2 contains minor changes and an extra remark. To appear in Bulletin of the London Mathematical Societ

Legal incapacitation in media access interdictions and the internet

In this paper I first situate the general norm in the Council of Europe countries on access to the internet by actors in the media in the context of key concerns related to the internet as a medium according to UNESCO’s Internet Study. Following on I provide analysis of media access to public events and places within the scope for protection according to recent sources from the Council of Europe’s political decision-making organs and the case-law of Human Rights Court. Further I delineate the scope for protection to digital free expression by a comparison of cases in the Netherlands on the internet as a means of access vis-à-vis a means of publication on social media and blogs. On this theme I focus on the particular motivations for protection of a wide-range of behavior related to the internet and specifics thereof in the parliamentary history of the Dutch law. Suggested by their wording, these political and judicial decisions limit the European Court’s general invocation of journalisms ethics when such is related to certain ‘ increased protections ’ to journalists. Some cases have - as instances of decisional law - given journalists/press the ‘ benefit of the doubt ’. I delineate how the Court has found the rights to seek, impart and receive up until recently, as well related to internet. Article 10 ECHR and other sources (eg. the Declarations and Recommendations of the decision-making organs of the Council of Europe) provide a scope for protection. We’re reminded that not only Article 10 defines what comes to the scope for protection, but also rights to observe protests (article 11 ECHR), eg. The case-law of the Netherlands has suggested that privacy intrusions are the most upcoming issue of media access in the field of communications and the internet. Injunctive relief with targeted, specifics for execution has be offered for certain heightened privacy protection. General laws on informational acts, such as beleaguering on the internet shouldn’t lead to denying a person of means of communication if no alternative, similar means are available. I related the analysis finally to the question why we should define professionalism as an ongoing and timely issue. The definition attempted by the Court / Council of Europe according to different sources of international standards has been related to the concept of ‘responsible journalism’, but the European Court can be seen to use a too simple view of what sources constitute its test(s) in a case

The role of affective information in the sociomoral development of preschool children

Early theories of social-cognitive development emphasized children\u27s unilateral respect for rules and authority (Piaget, 1932/65). Recently, however, children have been found to make conceptual distinctions among moral, conventional, and personal events. These distinctions are hypothesized to be related to the differential judgments that 6- to 8-year-old children have been shown to make about the emotional experience of others in these types of events (Arsenio & Ford, 1985). However, it is not known if preschoolers make similar distinctions between the affect of various participants in different event-types. The judgments of preschoolers regarding the affect of participants in sociomoral events were examined. Forty-two 3-, 4-, and 5-year-olds were presented with scenarios depicting different sociomoral event-types, and asked to assess the affective consequences of events for story participants. Results show that preschoolers differentiate type of affect between sociomoral events and between event participants. Potential implications are that young children are aware of the emotional consequences of events and that this knowledge plays a role in their sociomoral judgments

Exotic C*-algebras of geometric groups

We consider a new class of potentially exotic group C*-algebras $C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$ for $2<p<\infty$ when $G$ is either a noncommutative free group or the group $SL(2,\mathbb R)$, respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group $G$. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of $G$ in the left regular representation of $G$. Also we give a near solution to a 1978 conjecture of Cowling. This conjecture of Cowling states if $G$ is a Kunze-Stein group and $\pi$ is a unitary representation of $G$ with cyclic vector $\xi$ such that the map $$G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$$ belongs to $L^p(G)$ for some $2< p <\infty$, then $A_\pi\subseteq L^p(G)$. We show $B_\pi\subseteq L^{p+\epsilon}(G)$ for every $\epsilon>0$ (recall $A_\pi\subseteq B_\pi$)

Magnetic field generation in relativistic shocks - An early end of the exponential Weibel instability in electron-proton plasmas

We discuss magnetic field generation by the proton Weibel instability in relativistic shocks, a situation that applies to the external shocks in the fireball model for Gamma-ray Bursts, and possibly also to internal shocks. Our analytical estimates show that the linear phase of the instability ends well before it has converted a significant fraction of the energy in the proton beam into magnetic energy: the conversion efficiency is much smaller (of order m_e/m_p) in electron-proton plasmas than in pair plasmas. We find this estimate by modelling the plasma in the shock transition zone with a waterbag momentum distribution for the protons and with a background of hot electrons. For ultra-relativistic shocks we find that the wavelength of the most efficient mode for magnetic field generation equals the electron skin depth, that the relevant nonlinear stabilization mechanism is magnetic trapping, and that the presence of the hot electrons limits the typical magnetic field strength generated by this mode so that it does not depend on the energy content of the protons. We conclude that other processes than the linear Weibel instability must convert the free energy of the protons into magnetic fields.Comment: 7 pages, 3 figures, accepted for publication in A&