Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux; H. Radjavi; A. R. Sourour

Displaying similar documents to “Multiplicative maps that are close to an automorphism on algebras of linear transformations”

The Strong Anick Conjecture is true

Vesselin Drensky, Jie-Tai Yu (2007)

Journal of the European Mathematical Society

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Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra $K 〈 x, y, z 〉$ over a field $K$ of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of $K 〈 x, y, z 〉$ . In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We...

Automorphisms of central extensions of type I von Neumann algebras

Sergio Albeverio, Shavkat Ayupov, Karimbergen Kudaybergenov, Rauaj Djumamuratov (2011)

Studia Mathematica

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Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as $T = T_{a} \circ T_{ϕ}$ , where $T_{a} (x) = a x a^{- 1}$ is an inner automorphism implemented by an element a ∈ E(M), and $T_{ϕ}$ is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type $I_{\infty}$ then every band preserving...

Automorphisms of $(λ) / ℐ_{κ}$

Paul Larson, Paul McKenney (2016)

Fundamenta Mathematicae

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We study conditions on automorphisms of Boolean algebras of the form $(λ) / ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $(2^{κ}) / ℐ_{κ ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M A_{ℵ ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

If the [T,Id] automorphism is Bernoulli then the [T,Id] endomorphism is standard

Christopher Hoffman, Daniel Rudolph (2003)

Studia Mathematica

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For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and $[T, T^{- 1}]$ automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli...

Combinatorial mapping-torus, branched surfaces and free group automorphisms

François Gautero (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a $1$ -cocycle of a $2$ -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction...

Non-standard automorphisms of branched coverings of a disk and a sphere

Bronisław Wajnryb, Agnieszka Wiśniowska-Wajnryb (2012)

Fundamenta Mathematicae

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Let Y be a closed 2-dimensional disk or a 2-sphere. We consider a simple, d-sheeted branched covering π: X → Y. We fix a base point A₀ in Y (A₀ ∈ ∂Y if Y is a disk). We consider the homeomorphisms h of Y which fix ∂Y pointwise and lift to homeomorphisms ϕ of X-the automorphisms of π. We prove that if Y is a sphere then every such ϕ is isotopic by a fiber-preserving isotopy to an automorphism which fixes the fiber $π^{- 1} (A ₀)$ pointwise. If Y is a disk, we describe explicitly a small set of automorphisms...

On Automorphisms of the Affine Cremona Group

Hanspeter Kraft, Immanuel Stampfli (2013)

Annales de l’institut Fourier

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We show that every automorphism of the group $𝒢_{n} : = A u t (𝔸^{n})$ 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.

More on tie-points and homeomorphism in ℕ*

Alan Dow, Saharon Shelah (2009)

Fundamenta Mathematicae

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A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X = A ⋈_{x} B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique...

Lifts for semigroups of endomorphisms of an independence algebra

João Araújo (2006)

Colloquium Mathematicae

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For a universal algebra , let End() and Aut() denote, respectively, the endomorphism monoid and the automorphism group of . Let S be a semigroup and let T be a characteristic subsemigroup of S. We say that ϕ ∈ Aut(S) is a lift for ψ ∈ Aut(T) if ϕ|T = ψ. For ψ ∈ Aut(T) we denote by L(ψ) the set of lifts of ψ, that is, $L (ψ) = {ϕ \in A u t (S) | ϕ |}_{T} = ψ$ $.$ Let be an independence algebra of infinite rank and let S be a monoid of monomorphisms such that G = Aut() ≤ S ≤ End(). It is obvious that G is characteristic...

Automorphisms of the algebra of operators in $^{p}$ preserving conditioning

Ryszard Jajte (2010)

Colloquium Mathematicae

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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].

Coleman automorphisms of finite groups with a self-centralizing normal subgroup

Jinke Hai (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group with a normal subgroup $N$ such that $C_{G} (N) \leq N$ . It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

The automorphism group of the Lebesgue measure has no non-trivial subgroups of index $< 2^{ω}$

Maciej Malicki (2013)

Colloquium Mathematicae

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We show that the automorphism group Aut([0,1],λ) of the Lebesgue measure has no non-trivial subgroups of index $< 2^{ω}$ .

Discretized $C^{*}$ -Algebras

Carla Farsi, Neil Watling (2006)

Bollettino dell'Unione Matematica Italiana

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We define discretized canonical commutation relations associated to finite order automorphisms of discrete abelian groups. This generalizes the situation for rotation algebras and their finite order automorphisms. We also consider the almost Schrödinger operator associated to the given commutation relations.

The group of automorphisms of $L_{\infty}$ is algebraically reflexive

Félix Cabello Sánchez (2004)

Studia Mathematica

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

Нормальные автоморфизмы в многообразии $𝒩_{2} 𝒜$ . про- $p$ -группы

Ч.К. Гупта, Н.С. Романовский, Č. K. Gupta, N. S. Romanovskij, Č. K. Gupta, N. S. Romanovskij, Č. K. Gupta, N. S. Romanovskij (1996)

Algebra i Logika

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Infinitesimal CR automorphisms for a class of polynomial models

Martin Kolář, Francine Meylan (2017)

Archivum Mathematicum

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In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in $ℂ^{3}$ of the form $ℑ w = ℜ (P (z) \bar{Q (z)})$ , where $P$ and $Q$ are weighted homogeneous holomorphic polynomials in $z = (z_{1}, z_{2})$ . We classify such models according to their Lie algebra of infinitesimal CR automorphisms. We also give the first example of a non monomial model which admits a nonlinear rigid automorphism. ...

Artinian automorphisms of infinite groups

Antonella Leone (2006)

Bollettino dell'Unione Matematica Italiana

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An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_{1}>H_{2}>\cdots>H_{n}>\cdots$ of subgroups of G there exists a positive integer $m$ such that $(H_{n})^{a}=H_{n}$ for every $n\geq m$ . 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 $G$ .

Automorphisms of completely primary finite rings of characteristic p

Chiteng'a John Chikunji (2008)

Colloquium Mathematicae

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A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and $R / ≅ G F (p^{r})$ , the finite field of $p^{r}$ elements, for any prime p and any positive integer r.

The Banach-Lie Group of Lie Automorphisms of an $H^{*}$ -Algebra

Antonio J. Calderón Martín, Candido Martín González (2007)

Bollettino dell'Unione Matematica Italiana

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We study the Banach-Lie group $\operatorname{Aut}(A^{-})$ 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 $\operatorname{Aut}(A^{-})_{0}=\operatorname{Aut}(A)$ implying $\operatorname{Der}(A)=\operatorname{Der}(A^{-})$ . In the finite dimensional case $\operatorname{Aut}(A^{-})_{0}$ is a direct product of $\operatorname{Aut}(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.

Balls for the Kobayashi distance and extension of the automorphisms of strictly convex domains in $C^{n}$ with real analytic boundary

Andrea Iannuzzi (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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It is shown that given a bounded strictly convex domain $Ω$ in $C^{n}$ with real analitic boundary and a point $x_{0}$ in $Ω$ , there exists a larger bounded strictly convex domain $Ω^{'}$ with real analitic boundary, close as wished to $Ω$ , such that $Ω$ is a ball for the Kobayashi distance of $Ω^{'}$ with center $x_{0}$ . The result is applied to prove that if $Ω$ is not biholomorphic to the ball then any automorphism of $Ω$ extends to an automorphism of $Ω^{'}$ .

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

F. Ghahramani, J. McClure (1992)

Studia Mathematica

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

The Wells map for abelian extensions of 3-Lie algebras

Youjun Tan, Senrong Xu (2019)

Czechoslovak Mathematical Journal

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The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0 \to A ↪ L \overset{π}{\to} B \to 0$ of 3-Lie algebras to obtain obstruction classes in $H^{1} (B, A)$ for a pair of automorphisms in $Aut (A) \times Aut (B)$ to be inducible from an automorphism of $L$ . Application to free nilpotent 3-Lie algebras is discussed.