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Register a new userMaximal von Neumann subalgebras arising from maximal subgroups
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Yongle Jiang
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Ge asked the question whether $LF_{infty}$ can be embedded into $LF_2$ as a maximal subfactor. We answer it affirmatively by three different approaches, all containing the same key ingredient: the existence of maximal subgroups with infinite index. We also show that point stabilizer subgroups for every faithful, 4-transitive action on an infinite set give rise to maximal von Neumann subalgebras. Combining this with known results on constructing faithful, highly transitive actions, we get many maximal von Neumann subalgebras arising from maximal subgroups with infinite index.
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We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $mathbb{Z}^2 rtimes SL_2(mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.
In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of g and give explicit details on when it may be maximal in g.
We prove that $L(SL_2(textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(textbf{k}^2rtimes SL_2(textbf{k}))$ for $textbf{k}=mathbb{Q}$. Then we show how to modify the proof to handle $textbf{k}=mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(SL_2(textbf{k}))$ and $L^{infty}(Y)rtimes SL_2(textbf{k})$, where $SL_2(textbf{k})curvearrowright Y$ denotes the quotient of the algebraic action $SL_2(textbf{k})curvearrowright widehat{textbf{k}^2}$ by modding out the relation $phisim phi$, where $phi$, $phiin widehat{textbf{k}^2}$ and $phi(x, y):=phi(-x, -y)$ for all $(x, y)in textbf{k}^2$. As a by-product, we show $L(PSL_2(mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{infty}(Y)rtimes PSL_2(mathbb{Q})$; in particular, $PSL_2(mathbb{Q})curvearrowright Y$ is a prime action, i.e. it admits no non-trivial quotient actions.
We show that certain amenable subgroups inside $tilde{A}_2$-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann subalgebras. We also give a geometric proof that if $G$ is an acylindrically hyperbolic group, $H$ is an infinite amenable subgroup containing a loxodromic element, then $H<G$ is singular. Finally, we present (counter)examples to show both situations happen concerning maximal amenability of $LH$ inside $LG$ if $H$ does not contain loxodromic elements.
As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerups $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.
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