A short proof that ${\mathcal B}(L_1)$ is not amenable
Yemon Choi
备注：
v3: AMS-LaTeX, 10 pages. Incorporates referee's feedback; typos fixed and one reference added. Final version, to appear in Proc. Roy. Soc. Edinburgh Sect. A
发表信息：
Proc. Roy. Soc. Edinburgh Sect. A 151 (2021) , no. 6, 1758-1767
Non-amenability of ${\mathcal B}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for $E= \ell_p$ and $E=L_p$ for all $1\leq p<\infty$. However, the arguments are rather indirect: the proof for $L_1$ goes via non-amenability of $\ell^\infty({\mathcal K}(\ell_1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that ${\mathcal B}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on $L_1$, and shows that ${\mathcal B}(L_1)$ is not even approximately amenable.
In this paper, we study the notion of Connes amenability for a class of $I\times{I}$-upper triangular matrix algebra $UP(I,\mathcal{A})$, where $\mathcal{A}$ is a dual Banach algebra with a non-zero $wk^\ast$-continuous character and $I$ is a totally ordered set. For this purpose, we characterize the $ϕ$-Connes amenability of a dual Banach algebra $\mathcal{A}$ through the existence of a specified net in $\mathcal{A}\hat{\otimes}\mathcal{A}$, where $ϕ$ is a non-zero $wk^\ast$-continuous character. Using this, we show that $UP(I,\mathcal{A})$ is Connes amenable if and only if $I$ is singleton a…
Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets $(ξ_α)_α$ and $(μ_α)_α$ in $\mathcal{X}$ such that, for each $a\in\mathfrak{A}$, $\mathcal{D}(a)=\lim_α(a.ξ_α-μ_α.a)$. $\mathfrak{A}$ is called approximately semi-amenable if for every Banach $\mathfrak{A}$-bimodule $\mathcal{X}$, every $\mathcal{D}\in\mathcal{Z}^{1}(\mathfrak{A},\mathcal{X}^{*})$ is approximtely semi-inner. There are some Banach algebras which are appro…
An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.
This paper continues the investigation of Esslamzadeh and the first author which was begun in [7]. It is shown that homomorphic image of an approximately cyclic amenable Banach algebra is again approximately cyclic amenable. Equivalence of approximate cyclic amenability of a Banach algebra $\mathcal{A}$ and approximate cyclic amenability of $M_{n}(\mathcal{A})$ is proved. It is shown that under certain conditions the approximate cyclic amenability of second dual $\mathcal{A}^{**}$ implies the approximate cyclic amenability of $\mathcal{A}$.
We will extend earlier transference results of Neuwirth and Ricard from the context of noncommutative $L_p$-spaces associated with amenable groups to that of noncommutative $L_p$-spaces over crossed products of amenable and trace-preserving actions. Namely, if $T_m:L_p(\mathcal{L} G) \rightarrow L_p(\mathcal{L} G)$ is a completely bounded Fourier multiplier, where $\mathcal{L} G \subset \mathcal{B}(L_2 G)$ is the von Neumann algebra of $G$, we will see that $\mathrm{Id} \rtimes T_m: L_p(\mathcal{M} \rtimes_θG) \rightarrow L_p(\mathcal{M} \rtimes_θG)$ is also completely bounded and that $$ \| \…
Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a countable amenable group $G$. If the trace space $T(A)$ is a Bauer simplex and the action of $G$ on $\partial_eT(A)$ has finite orbits and Hausdorff orbit space, we show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, then these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2)…
We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.
This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely …
Let $G$ be an amenable group. We define and study an algebra $\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely…
Let $E$ be a Banach space, and $\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\mathcal B(E)$ goes back to Johnson's seminal memoir \cite{johnson} from 1972. We present the first general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space geometry of $E$, which imply that $\mathcal B(E)$ is non-amenable. We cover all spaces for which this is known so far (with the exception of one particular example), with much shorter proofs, such as $\ell_p$ for $p \in [1, \infty]$ and $c_0$, but also many new spaces:…
Suppose that ${\mathcal A}$ is an algebra, $σ,τ:{\mathcal A}\to{\mathcal A}$ are two linear mappings such that both $σ({\mathcal A})$ and $τ({\mathcal A})$ are subalgebras of ${\mathcal A}$ and ${\mathcal X}$ is a $\big(τ({\mathcal A}),σ({\mathcal A})\big)$-bimodule. A linear mapping $D:{\mathcal A}\to {\mathcal X}$ is called a $(σ,τ)$-derivation if $D(ab)=D(a)\cdotσ(b)+τ(a)\cdot D(b) (a,b\in {\mathcal A})$. A $(σ,τ)$-derivation $D$ is called a $(σ,τ)$-inner derivation if there exists an $x\in{\mathcal X}$ such that $D$ is of the form either $D_x^-(a)=x\cdot σ(a)-τ(a)\cdot x (a\in {\mathcal A}…
For a Banach algebra $ \mathcal{A} $, we introduce various approximate virtual diagonals such as approximate WAP-virtual diagonal and approximate virtual diagonal. For the enveloping dual Banach algebra $ F(\mathcal{A}) $ of $ \mathcal{A} $, we show that $ F(\mathcal{A}) $ is approximately Connes-amenable if and only if $ \mathcal{A} $ has an approximate WAP-virtual diagonal. Further, for a discrete group $ G $, we show that if the group algebra $ \ell^1(G) $ has an approximate WAP-virtual diagonal, then it has an approximate virtual diagonal.
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…
Let $\mathcal A$, $\mathcal B$ be Banach $\mathfrak A$-modules with compatible actions and $\mathcal M$ be a left Banach $\mathcal A$-$\mathfrak A$-module and a right Banach $\mathcal B$-$\mathfrak A$-module. In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$ (as an $\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of idempotents, the triangul…
We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann alg…
We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of emb…
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Let $G$ be a discrete group. Let $λ: G \to B(\ell_2(G),\ell_2(G))$ be the left regular representation. A function $\ph : G \to \comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of $\{λ(x),x \in G\}$ by $$\sum_{x \in G} f(x) λ(x) \to \sum_{x \in G} f(x) \ph(x) λ(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_λ^*(G)$ which is generated by $λ$ ($C_λ^*(G)$ is the closure of $K(G)$ in $B(\ell_2(G),\ell_2(G))$.) One of our main results gives a simple characterization of the functions $\ph$ such that …
Let $Γ\curvearrowright Ω$ be a measure-preserving action and $\mathcal{L} Γ\hookrightarrow L^\infty(Ω) \rtimes Γ$ the natural inclusion of the group von Neumann algebra into the crossed product. When $μ(Ω) = \infty$, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative $L^p$-spaces. Nevertheless, we show that when $Ω$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} Γ)$ into an ultrapower of $L^p(Ω\rtimes Γ)$ that intertwines Fourier multipliers and is $\mathcal{L} Γ$-bimodular. As a conse…
In this short note, we show that R. Thompson's group $F$ admits a normalish amenable subgroup, and that the standard copy of $F$ in R. Thompson's group $T$ is normalish in $T$. We further conjecture that if $F$ is non-amenable, then $T$ does not admit a normalish amenable subgroup, and therefore that the reduced $C^*$ algebra of $T$ is in fact simple in that case.
We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphe…
We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability.
As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group.
We outline a simple proof of Hulanicki's theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.
We define and analyse the $k$-directional short-time Fourier transform and its synthesis operator over Gelfand Shilov spaces $\mathcal S^α_β(\mathbb R^n)$ and $\mathcal S^α_β(\mathbb R^{k+n})$ respectively, and their duals. Also, we investigate directional regular sets and their complements - directional wave fronts, for elements of $\mathcal S^{\prime α}_α(\mathbb R^n)$.
We discuss the structural and topological properties of a general class of weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted $\mathcal{D}'_{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov spaces $\mathcal{K}\{M_p\}$. In particular, we characterize the sequences of weight functions $(M_p)_{p \in \mathbb{N}}$ for which the space of convolutors of $\mathcal{K}\{M_p\}$ is ultrabornological, thereby generalizing Grothendieck's classical result for the space $\mathcal{O}'_{C}$ of rapidly decreasing distributions. Our methods lead to the first direct p…
In this short note, further to Ng's study, we extend Bekka amenability and weak Bekka amenability to general locally compact quantum groups. We generalize some Ng's results to the general case. In particular, we show that, a locally compact quantum group $\mathbb{G}$ is co-amenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka amenable, and $\mathbb{G}$ is amenable if and only if its dual quantum group's fundamental multiplicative unitary $W_{\widehat{\mathbb{G}}}$ is weakly Bekka amenable.
We give some new results related to the directional short-time Fourier transform (DSTFT) and extend them on the spaces $\mathcal K_{1}(\mathbb R^{n})$ and $\mathcal K_{1}({\mathbb R})\widehat{\otimes}\mathcal U(\mathbb C^n)$ and their duals. Then, we define multi-directional STFT and, for tempered distributions, directional regular sets and their complements, directional wave fronts. Different windows with mild conditions on their support show the invariance of these notions related to window functions. Smoothness of $f$ follows from the assumptions of the directional regularity in any directi…
