Local Automorphisms Research Articles

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ n+1 admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ n+1 with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group.

Local automorphisms of the Hilbert ball

Every holomorphic mapping which takes a piece of the boundary of the unit ball in complex Hilbert space into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gâteaux-holomorphic.

LOCAL AUTOMORPHISMS OF OPERATOR ALGEBRAS

A not necessarily continuous, linear or multiplicative function $\theta$ from an algebra $\cal A$ into itself is called a local automorphism if $\theta$ agrees with an automorphism of $\cal A$ at each point in $\cal A$. In this paper, we study the question when a local automorphism of a $C$*-algebra, or a W*-algebra, is an automorphism.

Finite Jet Determination of Local CR Automorphisms through Resolution of Degeneracies

Let MC N be a connected real-analytic hypersurface whose Levi form is nonde- generate at some point. We prove that for every point p 2 M, there exists an integer k = k(M, p) such that germs at p of local real-analytic CR automorphisms of M are uniquely determined by their k-jets (at p). To prove this result we develop a new technique that can be seen as a resolution of the degeneracies of M. This procedure consists of blowing up M near an arbitrary point p 2 M regardless of its minimality or nonminimality; then, thanks to the blow-up, the original problem can be reduced to an analogous one for a very special class of nonminimal hypersurfaces for which one may use known techniques to prove the finite jet determination property of its CR automorphisms.

2-Local isometries and 2-local automorphisms on uniform algebras

2-Local isometries and 2-local automorphisms on uniform algebras

Local symmetries of finite type hypersurfaces in ℂ2

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in ℂ2. It is also proved that, with the exception of hypersurfaces of the form v = |z| k , local automorphisms are always determined by their 1-jets. Using this result, the second part describes special normal forms which by an additional normalization eliminate the nonlinear symmetries of the model and allows to decide effectively about local equivalence of two hypersurfaces given in this normal form.

A NOTE ON 2-LOCAL MAPS

AbstractThe aim of this note is to characterize 2-local automorphisms and derivations on matrix rings over finite-dimensional division rings.

A note on local automorphisms

Let H be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of ℬ(H), the algebra of all bounded linear operators on a Hilbert space H, is an automorphism.

On the stability group of CR manifolds

For any essentially finite minimal real-analytic generic submanifold M ⊂ C N , N ⩾ 2 , we show that for every point p ∈ M the local real-analytic CR automorphisms of M fixing p can be parametrized real-analytically by their ℓ = ℓ ( p ) jets at p. As an application, we derive a Lie group structure for the stability group Aut ( M , p ) . We also show that the order ℓ = ℓ ( p ) of the jet space in which the group Aut ( M , p ) embeds can be chosen to depend upper-semicontinuously on p. This yields that given any compact real-analytic minimal CR submanifold M in C N , there exists an integer k depending only on M such that for every point p ∈ M local CR diffeomorphisms mapping a neighbourhood of p in M into another real-analytic CR submanifold in C N with the same CR dimension as that of M are uniquely determined by their k-jet at p. To cite this article: B. Lamel, N. Mir, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface M ⊂ C N M\subset \mathbb {C}^N , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p ∈ M p\in M the local real-analytic CR automorphisms of M M fixing p p can be parametrized real-analytically by their ℓ p \ell _p jets at p p . As a direct application, we derive a Lie group structure for the topological group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) . Furthermore, we also show that the order ℓ p \ell _p of the jet space in which the group Aut ⁡ ( M , p ) \operatorname {Aut}(M,p) embeds can be chosen to depend upper-semicontinuously on p p . As a first consequence, it follows that given any compact real-analytic hypersurface M M in C N \mathbb {C}^N , there exists an integer k k depending only on M M such that for every point p ∈ M p\in M germs at p p of CR diffeomorphisms mapping M M into another real-analytic hypersurface in C N \mathbb {C}^N are uniquely determined by their k k -jet at that point. Another consequence is the following boundary version of H. Cartan’s uniqueness theorem: given any bounded domain Ω \Omega with smooth real-analytic boundary, there exists an integer k k depending only on ∂ Ω \partial \Omega such that if H : Ω → Ω H\colon \Omega \to \Omega is a proper holomorphic mapping extending smoothly up to ∂ Ω \partial \Omega near some point p ∈ ∂ Ω p\in \partial \Omega with the same k k -jet at p p with that of the identity mapping, then necessarily H = Id H=\textrm {Id} . Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

KMS States and the Chemical Potential for Disordered Systems

We extend the theory of the chemical potential associated to a compact separable gauge group, to the case of disordered quantum systems. This is done in the framework of operator algebras. Among the other results, we show that the chemical potential does not depend on the disorder. The situation of the n–torus is treated in some detail. Indeed, the chemical potential is intrinsically defined in terms of the direct integral decomposition of the Connes–Radon–Nikodym cocycle associated to the (normal extension of the) KMS state ω and its trasforms ω○ρ by the (localized) automorphisms ρ of the observable algebra, carrying the Abelian charges of the model under consideration. This description parallels the analogous one relative to the usual (i.e. non disordered) quantum models.

Local isometries on spaces of continuous functions

Let X and Y be Banach spaces and S a subset of the space of (linear, continuous) operators from X to Y . We say that an operator T belongs locally to S if for every x ∈ X there is S ∈ S, possibly depending on x, such that Tx = Sx. ‘Pointwise’ should be better than ‘locally’, but we have followed tradition. If each operator that belongs locally to S belongs in fact to S we say that S is algebraically reflexive. When Y = X and S = Iso(X) is the group of isometries of X we say that T is a local isometry of X. (In this paper ‘isometry’ means linear surjective isometry.) Similarly, a local automorphism of a Banach algebra is an operator which agrees at every point with some automorphism. Also, we will consider approximate local isometries and automorphisms. These are operators having with the following property: given x ∈ X and e > 0, there is an isometry (respectively, an automorphism) S of X such that ‖Tx− Sx‖ < e. The study of local isometries and automorphisms of Banach algebras spurred a considerable interest in recent years (see the bibliography of the dissertation [16]). In this paper we deal with local isometries and automorphisms of the algebras C0(L). As usual, we write C0(L) or C K 0 (L) for the Banach algebra of all continuous K-valued functions on the locally compact space L vanishing at infinity, where K is either C or R. If L is compact the subscript will be omitted. By the Banach-Stone theorem, if T is local isometry of C0(L), then for each f there are a homeomorphism φ of L and a continuous unitary u : L→ K such that Tf = u(f ◦ φ).

Maps on idempotent matrices over division rings

Let D be an arbitrary division ring and P n ( D ) the set of all n × n idempotent matrices over D . Under some mild conditions, we give a complete description of maps on P n ( D ) that preserve either commutativity, or order, or orthogonality. We give examples showing that our assumptions cannot be relaxed much further. As an application, we will prove a quaternionic analogue of Ovchinnikov's result that is important in quantum mechanics. Other applications of our theorems include results on automorphisms of operator and matrix semigroups, local automorphisms, linear preserver problems and geometry of matrices and Grassmannians.

Graph-theoretical characterization of periodicity in crystallographic nets and other infinite graphs

Local automorphisms in infinite graphs are defined as those automorphisms for which the distance (in the graph-theoretical sense) between any vertex and its image possesses an upper bound. Abelian subgroups of direction-preserving local automorphisms without fixed point, so-called shift groups, are used to determine the quotient graph of infinite graphs. It is shown that the shift group, the closest topological analogue to a translation group in crystal structures, is isomorphic to the quotient group C/C(0) of the cycle space C of the quotient graph by some subgroup C(0), its kernel. As a main consequence, the isomorphism class of nets can be determined directly from their labeled quotient graph, without having recourse to any embedding. A general method is formulated and illustrated in the case of cristobalite and moganite structures. Application to carbon and other kinds of nanotubes is also described.

Local automorphisms and derivations on certain 𝐶*-algebras

It is shown that continuous 2 2 -local derivations on AF \operatorname {AF} C ∗ C^* -algebras are derivations and surjective 2 2 -local *-automorphisms on prime C ∗ C^* -algebras or on C ∗ C^* -algebras such that the identity element is properly infinite are *-automorphisms.

Local automorphisms of nest subalgebras of factor von Neumann algebras

In this paper, it is proved that every surjective weakly continuous linear local automorphism of nest subalgebras with non-trivial nests of factor von Neumann algebras is an automorphism; and that every surjective linear 2-local automorphism of nest algebras with non-trivial nests of factor von Neumann algebras is an automorphism.

Algebras of infinitesimal CR automorphisms

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional. The authors investigate some formal properties of CR algebras and construct some fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the E. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds. One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three parts'': totally real, totally complex and weakly nondegenerate CR algebras (Theorems 5.3 and 5.4). There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations.

Automorphisms of Hilbert space effect algebras

We consider bijections of the Hilbert space effect algebra that preserve the algebraic structures in one direction and have some other properties. It is shown that if ϕ : E ( H ) → E ( H ) is conditionally multiplicative and conditionally additive, then ϕ is implemented by a unitary or antiunitary operator on H . We also show that 2-local ortho-order automorphisms on E ( H ) are of the same form if dim H ⩾ 3 .

Dynamics of Local Automorphisms of Embedded CR-Manifolds

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not. Let M be a real-analytic Levi-nondegenerate hypersurface in some neighborhood U of the origin in C , n ≥ 1 . Denote by Aut0 M the group of local diffeomorphisms of M , that is, the group of germs of biholomorphisms taking M locally into itself and preserving the origin. A point q = 0 (treated as its own radius-vector −→ 0q) is said to be contractible if there is a sequence of local automorphisms Φn ∈ Aut0 M such that

Nonalgebraizable real analytic tubes in C n

We give necessary conditions for certain real analytic tube generic submanifolds in ℂn to be locally algebraizable. As an application, we exhibit families of real analytic non locally algebraizable tube generic submanifolds in ℂn. During the proof, we show that the local CR automorphism group of a minimal, finitely nondegenerate real algebraic generic submanifold is a real algebraic local Lie group. We may state one of the main results as follows. Let M be a real analytic hypersurface tube in ℂn passing through the origin, having a defining equation of the form v= φ(y), where (z,w)=(x+iy,u+iv)∈Open image in new windown−1×Open image in new window. Assume that M is Levi nondegenerate at the origin and that the real Lie algebra of local infinitesimal CR automorphisms of M is of minimal possible dimension n, i.e. generated by the real parts of the holomorphic vector fields ∂z1,...,∂z n−1,∂w. Then M is locally algebraizable only if every second derivative ∂2yky lφ; is an algebraic function of the collection of first derivatives ∂y1φ,⋯,∂ymφ.