Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups
Pekka Salmi
备注：
21 pages. Final version. Minor edits to version 2
发表信息：
Pekka Salmi, Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups, Journal of Functional Analysis 261 (2011) 1-24
We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $\mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(\mathbb{G})$. We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group $G$ and non-zero, invariant C*-subalgebras of the group C*-algebra $C^*(G)$.
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.
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 rem…
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We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $LΛ\subset LΓ$ arising from amenable subgroups $Λ$ of discrete countable groups $Γ$. The criterion is expressed in terms of $Λ$-invariant measures on some compact $Γ$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C*-algebra.
Given an irreducible lattice $Γ$ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $Γ$-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal{L}(Γ)$, and for the $Γ$-invariant unital $C^*$-subalgebras of the reduced group $C^*$-algebra $C^*_{\rm red}(Γ)$. We use these results to show that: (i) every $Γ$-invariant von Neumann subalgebra of $\mathcal{L}(Γ)$ is generated by a normal subgroup; and (ii) given a non-amenable unitary representation $π$ of $Γ$, every $Γ$-equivariant conditional expectation on $C^*_π(Γ)$ is the canonical cond…
It is a wide open problem to give an intrinsic criterion for a II_1 factor $M$ to admit a Cartan subalgebra $A$. When $A \subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is "simple" in the sense that the left and right action of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A II_1 factor $M$ that admits such a subalgebra $A$ is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II_1 factor $M$ to be s-thin and left open the question whether all s-thin II_1 factors admit a Cartan subalgebra. We answer this question negatively by construc…
We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$ factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.
By classifying $S$-maximal amenable subgroups of algebraic groups over a global field of characteristic zero, we obtain a complete classification of maximal amenable subgroups up to commensurability in the respective arithmetic groups. Futhermore, we prove that these commensurably maximal amenable subgroups are singular and therefore give rise to maximal amenable von Neumann subalgebras.
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Three separation properties for a closed subgroup $H$ of a locally compact group $G$ are studied: (1) the existence of a bounded approximate indicator for $H$, (2) the existence of a completely bounded invariant projection of $VN\left(G\right)$ onto $VN_{H}\left(G\right)$, and (3) the approximability of the characteristic function $χ_{H}$ by functions in $M_{cb}A\left(G\right)$ with respect to the weak$^{*}$ topology of $M_{cb}A\left(G_{d}\right)$. We show that the $H$-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for $H$ and…
We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
We say that a countable discrete group $Γ$ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every $Γ$- invariant von Neumann subalgebra $\mathcal{M}$ in $L(Γ)$ is of the form $L(Λ)$ for some normal subgroup $Λ\lhd Γ$. We show many ``negatively curved" groups, including all torsion free non-amenable hyperbolic groups and torsion free groups with positive first $L^2$-Betti number under a mild assumption, and certain finite direct product of them have this property. We also discuss whether the torsion-free assumption can be relaxed.
We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.
Using an interplay between geometric methods in group theory and soft von Neuman algebraic techniques we prove that for any icc, acylindrically hyperbolic group $Γ$ its von Neumann algebra $L(Γ)$ satisfies the so-called ISR property: \emph{any von Neumann subalgebra $N\subseteq L(Γ)$ that is normalized by all group elements in $Γ$ is of the form $N= L(Σ)$ for a normal subgroup $Σ\lhd Γ$.} In particular, this applies to all groups $Γ$ in each of the following classes: all icc (relatively) hyperbolic groups, most mapping class groups of surfaces, all outer automorphisms of free groups with at le…
For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor fro…
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