Gabor Elek
We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.
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We prove that every amenable group of cohomological dimension two whose integral group ring is a domain is solvable and investigate certain homological finiteness properties of groups that satisfy the analytic zero divisor conjecture and act on an acyclic CW-complex with amenable stabilisers.
There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, which supporting the conjecture.
A topological group $G$ is B-amenable if and only if every continuous affine action of $G$ on a bounded convex subset of a locally convex space has an approximate fixed point. Similar results hold more generally for slightly uniformly continuous semigroup actions.
Suppose that S is a left amenable semitopological semigroup. We prove that if ${T_{t}: t \in S}$ is a uniformly k-Lipschitzian semigroup on a bounded closed and convex subset C of a Hilbert space and $k<\sqrt{2}$, then the set of fixed points of this semigroup is a Hölder continuous retract of C. This gives a qualitative complement to the Ishihara-Takahashi fixed point existence theorem.
In this note we prove that in the case of finitely generated amenable groups the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.
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 give a counterexample to a conjecture of S.E. Morris by showing that there is a compact plane set X such that R(X) has no non-zero, bounded point derivations but such that R(X) is not weakly amenable. We also give an example of a separable uniform algebra A such that every maximal ideal of A has a bounded approximate identity but such that A is not weakly amenable.
We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison's conjecture on completely bounded homomorphisms holds for the algebra if the associated quotient is abelian. We also prove that injective completely bounded representations of the algebra are similar to complete isometries. The main motivating example for these investigations is the recent construction of Choi, Farah and Ozawa of an amenable operator algebra that is not similar to a $C^*$-algebra, and we show how it fits i…
We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of…
The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent w…
We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenab…
Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain numbe…
A locally compact group $G$ is called Hermitian if the spectrum $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called quasi-Hermitian if $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of "spectral interpolation of triple Banach $*$-algebr…
We prove that the $L^1$-algebra of any non-Kac type compact quantum group does not satisfy operator biflatness. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counter example to the conjecture that $L^1(\Gb)$ is operator amenable if and only if $\Gb$ is amenable and co-amenable for any locally compact quantum group $\Gb$. The result also implies that the $L^1$-algebra of a locally compact quantum group is operator biprojective if and only if $\Gb$ is compact and of Kac type.
Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G and in the speci…
Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.
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 $A$ 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 give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Barth…
Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.
Brennan's conjecture holds for semigroups of holomorphic functions
本文针对全纯函数连续半群这一特殊情形，结合经典复分析技术与Békollé-Bonami权和积分算子谱的近期结果，给出了Brennan猜想的一个简短证明。
Crouzeix's conjecture for classes of matrices
该研究通过构造矩阵类、揭示循环性与猜想的新联系，并研究对称矩阵情形，为证明Crouzeix猜想提供了新途径和开放问题。
Amenability for unitary groups of simple monotracial C*-algebras
本文证明单位正则无限核C*-代数的等距半群右可 Amenability，且迹 GNS 表示超有限的单位单迹C*-代数的酉群在弱拓扑下斜可 Amenability，部分解决相关猜想与问题。
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