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Register a new userAmenability of semigroups and common multiples in $ell^1_+$
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Authors
Tobias Fritz
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In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $ell^1_+(S)$ have a common nonzero right multiple, where $ell^1_+(S)$ is the positive part of the Banach algebra $ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.
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We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also F{o}lners type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no F{o}lner 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 natural generalization of the uniform Roe algebra of a group, which we denote by $mathcal{R}_X$. We show that the following notions are then equivalent: (1) $X$ is domain measurable; (2) $X$ is not paradoxical; (3) $X$ satisfies the domain F{o}lner condition; (4) there is an algebraically amenable dense *-subalgebra of $mathcal{R}_X$; (5) $mathcal{R}_X$ has an amenable trace; (6) $mathcal{R}_X$ is not properly infinite and (7) $[0] ot=[1]$ in the $K_0$-group of $mathcal{R}_X$. We also show that any tracial state on $mathcal{R}_X$ is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of $X$. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of $C_r^*left(Xright)$ implies the amenability of $X$. The converse implication is false.
Johnsons characterization of amenable groups states that a discrete group $Gamma$ is amenable if and only if $H_b^{n geq 1}(Gamma; V) = 0$ for all dual normed $mathbb{R}[Gamma]$-modules $V$. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective homomorphism $phi colon Gamma to K$ has amenable kernel $H$ if and only if the induced inflation map $H^bullet_b(K; V^H) to H^bullet_b(Gamma; V)$ is an isometric isomorphism for every dual normed $mathbb{R}[Gamma]$-module $V$. In addition, we obtain an analogous characterization for the (smaller) class of surjective homomorphisms $phi colon Gamma to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all normed $mathbb{R}[Gamma]$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms $phi colon Gamma to K$, for which the restriction maps in bounded cohomology $H^bullet_b(K; V) to H^bullet_b(Gamma; phi^{-1}V)$ are isomorphisms for suitable dual normed $mathbb{R}[K]$-module $V$. We then extend the first and third results to spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.
Rajchman measures of locally compact Abelian groups are studied for almost a century now, and they play an important role in the study of trigonometric series. Eymards influential work allowed generalizing these measures to the case of emph{non-Abelian} locally compact groups $G$. The Rajchman algebra of $G$, which we denote by $B_0(G)$, is the set of all elements of the Fourier-Stieltjes algebra that vanish at infinity. In the present article, we characterize the locally compact groups that have amenable Rajchman algebras. We show that $B_0(G)$ is amenable if and only if $G$ is compact and almost Abelian. On the other extreme, we present many examples of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which $B_0(G)$ fails to even have an approximate identity.
We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelsons space.
Associated to a nonzero homomorphism $varphi$ of a Banach algebra $A$, we regard special functionals, say $m_varphi$, on certain subspaces of $A^ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_varphi$ on $A^ast$; and, in addition we expatiate the case that if a functional $m_varphi$ is unique, then $m_varphi$ belongs to the topological center of the bidual algebra $A^{astast}$. An example of a function algebra, surprisingly, contradicts a conjecture that a Banach algebra $A$ is amenable if $A$ is $varphi$-amenable in every character $varphi$ and if functionals $m_varphi$ associated to the characters $varphi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras.
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