Narrow Results

Let $𝔘$ be a $\varphi$-Johnson amenable Banach algebra in which $\varphi$ is a nonzero multiplicative linear functional on $𝔘$. Suppose that $X$ is a Banach $𝔘$-bimodule such that $a.x=\varphi (a)x$ for all $a\in 𝔘$, $x\in X$ or $x.a=\varphi (a)x$ for all $a\in 𝔘$, $x\in X$. We show that every continuous Jordan derivation from $𝔘$ to $X$ is a derivation, and every continuous Lie derivation from $𝔘$ to $X$ decomposed into the sum of a continuous derivation and a continuous center-valued trace. Then we apply our results for character amenable Banach algebras and amenable Banach algebras. We also provide some results about $\varphi$-Johnson amenability, especially we give some conditions equivalent to $\varphi$-Johnson amenability.

We prove that any countable sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group can be embedded into a finitely generated sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group. We also prove an analogous result that any countable dimensional linearly sofic Lie algebra can be embedded into a finitely generated linearly sofic Lie algebra. In the course of proving this result, we prove that, over a field of characteristic 0, every extension of a linearly sofic Lie algebra by a Lie algebra with amenable universal enveloping algebra is linearly sofic.

Results in $C^{\ast }$ algebras, of Matte Bon and Le Boudec, and of Haagerup and Olesen, apply to the R. Thompson groups $F\le T\le V$. These results together show that $F$ is non-amenable if and only if $T$ has a simple reduced $C^{\ast }$-algebra.

In further investigations into the structure of $C^{\ast }$-algebras, Breuillard, Kalantar, Kennedy, and Ozawa introduce the notion of a normalish subgroup of a group $G$. They show that if a group $G$ admits no non-trivial finite normal subgroups and no normalish amenable subgroups then it has a simple reduced $C^{\ast }$-algebra. Our chief result concerns the R. Thompson groups $F<T<V$; we show that there is an elementary amenable group $E<F$ [where here, $E\cong \dots )\wr \mathbb{Z})\wr \mathbb{Z})\wr \mathbb{Z}$] with $E$ normalish in $V$.

The proof given uses a natural partial action of the group $V$ on a regular language determined by a synchronising automaton in order to verify a certain stability condition: once again highlighting the existence of interesting intersections of the theory of $V$ with various forms of formal language theory.

A CW-complex X is called a [G,m]-complex if X is an m-dimensional complex with π₁(X) ≅ G and the universal cover is (m - 1)-connected. We show that if G has an infinite amenable normal subgroup, then the asphericity of a [G,m]-complex X is equivalent to the vanishing of L²-Euler characteristic of . This result corresponds to a generalization and a variation of earlier several works. Also, we show that the L²-Betti numbers of a group which belongs to the class of groups K𝔉 eventually vanish. As a byproduct, we give an example of a group which belongs to the class of groups H𝔉 but does not belong to the class of groups K𝔉.

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci.110(12) (2013) 4524–4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn.10(1) (2016) 177–200]. These groups do not contain non-abelian free subgroups. In this paper, we prove that the Tarski number of all groups in both families is at most 25. In particular, we demonstrate the existence of a paradoxical decomposition with 25 pieces.

Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise $PS{L}_2(\mathbb{Z})$ homeomorphisms of $ℝ$ with rational breakpoints and an affine map that is a not an integer translation.

We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine–Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible actions of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of ${𝒜}⋊𝔄$ and give some necessary and sufficient conditions for injectivity, projectivity and flatness of ${𝒜}⋊𝔄$.

We study the concept of α-amenability of commutative hypergroups K. We establish several characterizations of α-amenability by combining results of Bloom, Heyer, and Filbir where we add the Glicksberg-Reiter property. In addition, as examples compact hypergroups and discrete polynomial hypergroups on ℕ₀ are discussed.