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1
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A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras

100%
Zhong Huang S.
Studia Mathematica
|
1999
|
tom 132
|
nr 1
37-69
EN
Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function $φ_a(t):=φ(α_t a)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum $σ_w*(φ_a)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define $Ʌ_φ^a$ to be the union of all sets $σ_w*(φ_a)$ where a ∈ A, and $Λ_α$ to be the closure of the union of all sets $Ʌ_φ^a$ where φ ∈ ∆(A), and call $Λ_α$ the unitary spectrum of α. Starting by showing that the closure of $Ʌ_φ^a$ (for fixed φ ∈ ∆(A)) is a subsemigroup of Ĝ we characterize the structure properties of the group representation α such as norm continuity, growth and existence of non-trivial invariant subspaces through its unitary spectrum $Λ_α.$ For an automorphism T of a semisimple commutative Banach algebra A we consider the group representation T: ℤ → Aut (A) defined by $T_n:=T^n$ for all n ∈ ℤ. It is shown that $Λ_T=σ(T)∩𝕋$, where σ(T) is the spectrum of T and 𝕋 is the unit circle. From this fact we give an easy proof of the Kamowitz-Scheinberg theorem which asserts that the spectrum σ(T) either contains 𝕋 or is a finite union of finite subgroups of 𝕋.
2
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The set of automorphisms of B(H) is topologically reflexive in B(B(H))

100%
Molnár L.
Studia Mathematica
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1997
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tom 122
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nr 2
183-193
EN
The aim of this paper is to prove the statement announced in the title which can be reformulated in the following way. Let H be a separable infinite-dimensional Hilbert space and let Φ: B(H) → B(H) be a continuous linear mapping with the property that for every A ∈ B(H) there exists a sequence $(Φ_n)$ of automorphisms of B(H) (depending on A) such that $Φ(A)= lim_n Φ_n(A)$. Then Φ is an automorphism. Moreover, a similar statement holds for the set of all surjective isometries of B(H).
3
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A characterization of linear automorphisms of the Euclidean ball

88%
Hamada H., Hamada H.
Annales Polonici Mathematici
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1999
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tom 72
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nr 1
79-85
EN
Let B be the open unit ball for a norm on $ℂ^n$. Let f:B → B be a holomorphic map with f(0) = 0. We consider a condition implying that f is linear on $ℂ^n$. Moreover, in the case of the Euclidean ball 𝔹, we show that f is a linear automorphism of 𝔹 under this condition.
4
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On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

51%
Tyszkowska E.
Open Mathematics
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2003
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tom 1
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nr 2
208-220
EN
The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p n are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most
5
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Lie algebraic characterization of manifolds

51%
Grabowski J., Poncin N.
Open Mathematics
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2004
|
tom 2
|
nr 5
811-825
EN
Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.
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