We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{\infty }\left(\mu \right)$ for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of $L_{\infty }\left(\mu \right)$ is [algebraically] reflexive if and only if $L_{\infty }\left(\mu \right)$ is *-isomorphic to $L_{\infty }[0,1]$. For purely atomic measures, we show that the group of automorphisms (or isometries) of ${\ell }_{\infty }\left(\Gamma \right)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism...

An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_1>H_2>\mathrm{\cdots }>H_n>\mathrm{\cdots }$ of subgroups of G there exists a positive integer $m$ such that ${(H_n)}^a=H_n$ for every $n\ge m$. In this paper we show that in many cases the group of all artinian automorphisms of $G$ coincides with the group of all power automorphisms of $G$.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L{¹}_{loc}$ of locally integrable functions on the half-line ${ℝ}^{+}$. We show, among other things, that every automorphism θ of $L{¹}_{loc}$ is of the form $\theta ={\phi }_a{e}^{\lambda X}{e}^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\phi }_a$ is the dilation operator $\left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)$ ($f\in L{¹}_{loc}$, $x\in {ℝ}^{+}$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence...

In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $\alpha$ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi :G\to G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{\text{'}\text{'}}$ is included in the centre of $H$ and $C_H\left({\alpha }^2\right)$ is abelian, both $C_G\left({\alpha }^2\right)$ and $G/[G,{\alpha }^2]$ are abelian-by-finite. These results extend recent and classical results in...

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all...

Let α be an isometric automorphism of the algebra ${}_p$ of bounded linear operators in ${}^p[0,1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

We show that every automorphism of the group ${𝒢}_n:=Aut\left({𝔸}^n\right)$ of polynomial automorphisms of complex affine $n$-space ${𝔸}^n={ℂ}^n$ is inner up to field automorphisms when restricted to the subgroup $T{𝒢}_n$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T{𝒢}_2={𝒢}_2$. The methods are different, based on arguments from algebraic group actions.

We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$-point on $\omega$, and show that these ultrafilters exist generically under $𝔠=𝔡$. This improves the known existence result of Ketonen [On the existence of $P$-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

We study the Banach-Lie group $Aut(A^{-})$ of Lie automorphisms of a complex associative $H^{*}$-algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$, are obtained. For a topologically simple $A$, in the infinite-dimensional case we have $Aut{(A^{-})}_0=Aut(A)$ implying $Der(A)=Der(A^{-})$. In the finite dimensional case $Aut{(A^{-})}_0$ is a direct product of $Aut(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.

We study the group $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ of tame automorphisms of a smooth affine $3$-dimensional quadric, which we can view as the underlying variety of ${\mathrm{SL}}_2\left(ℂ\right)$. We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $\mathrm{CAT}\left(0\right)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ is linearizable, and that $\mathrm{Tame}\left({\mathrm{SL}}_2\right)$ satisfies the Tits alternative.

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$ describing properties of Dense(ℚ). These invariants satisfy ${}_{\mathbb{Q}}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$.WecomparethemwiththeiranaloguesinthewellstudiedBooleanalgebra\left(\omega \right)/fin.Weshowthat$ℚ = p$,$ℚ = t$and$ℚ = i$,whereas$ℚ > h$and$ℚ > r$arebothshowntoberelativelyconsistentwithZFC.Wealsoinvestigatecombinatoricsoftheidealnwdofnowheredensesubsetsof,\mathbb{Q}.Inparticular,weshowthat$non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮*$..................................................................................................................... 5 § 3. $𝒮*$-algebras.....................................................................................................................

Let $R$ be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$, $G$ are generalized skew derivations of $R$ with the same associated automorphism $\alpha$, and $p(x_1,...,x_n)$ is a non-central polynomial over $C$ such that $$\left[F\right(x),\alpha (y\left)\right]=G\left(\right[x,y\left]\right)$$ for all $x,y\in \{p(r_1,...,r_n):r_1,...,r_n\in R\}$. Then there exists $\lambda \in C$ such that $F\left(x\right)=G\left(x\right)=\lambda \alpha \left(x\right)$ for all $x\in R$.

We study questions related to exceptional sets of pluri-Green potentials $V_{\mu }$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{\mu }$ are defined by $V_{\mu }\left(z\right)={\int }_{B}log(1/|{\varphi }_z\left(w\right)\left|\right)d\mu \left(w\right)$, where for a fixed z ∈ B, ${\varphi }_z$ denotes the holomorphic automorphism of B satisfying ${\varphi }_z\left(0\right)=z$, ${\varphi }_z\left(z\right)=0$ and $\left({\varphi }_z\circ {\varphi }_z\right)\left(w\right)=w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of...

We show that given infinite sets $X,Y$ and a function $f:X\to Y$ which is onto and $n$-to-one for some $n\in \mathbb{N}$, the preimage of any ultrafilter $ℱ$ of $Y$ under $f$ extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model $ℳ$ with a set of atoms $A$ and a finite-to-one onto function $f:A\to \omega$ such that for each free ultrafilter of $\omega$ its preimage under $f$ does not extend to an ultrafilter. In addition, we show that in $ℳ$ there exists an ultrafilter compact...

The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having $4n+6$ vertices and automorphism group cyclic of order $4n$, $n\ge 1$. As a special case we get graphs with $2^k+6$ vertices and cyclic automorphism groups of order $2^k$. It can revive interest in related problems.

Suppose $A$ is a separable unital $𝒞\left(X\right)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_1$-injective $C^{*}$-algebra $𝒟$. We show that $A$ and $𝒞\left(X\right)\otimes 𝒟$ are isomorphic as $𝒞\left(X\right)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

Let $k$ be an algebraically closed field of characteristic $p\&gt;0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

We investigate the question whether a system ${\left(E_i\right)}_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem ${\left(E_j\right)}_{j\in J}$ with $\left|J\right|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c<\left|I\right|$. Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $\left|I\right|={\aleph }_0$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le {\aleph }_0$ is ‘No...