Local Automorphisms Research Articles

Nilpotent local automorphisms of an independence algebra

Let A be a strong independence algebra of infinite rank m. Let ℒ(A) be the inverse monoid of all local automorphisms of A. The Baer–Levi semigroup B over the algebra A is defined to be the subsemigroup of ℒ(A) consisting of all the elements α with dom α = A and corank im α = m. Let Km be the subsemigroup of ℒ(A) generated by B−1B. Then Km is inverse and it is generated (as a semigroup) by N2, the subset of ℒ(A) consisting of all nilpotent elements of index 2. The 2-nilpotent depth, Δ2(Km) of Km is defined to be the smallest positive integer t such that Km = N2∪…∪(N2)t. In fact, Δ2(Km) is either 2 or 3 and a criterion is found which distinguishes between the two cases.If N denotes the set of all nilpotents in ℒ(A), then the subsemigroup generated by N is also Km. In fact, Km is proved to be exactly N2.

Local Derivations, Automorphisms and Commutativity Preserving Maps on $\mathscr L^+(\mathscr D)$

Local Derivations, Automorphisms and Commutativity Preserving Maps on $\mathscr L^+(\mathscr D)$

The Dirac oscillator and local automorphism invariance

The Dirac oscillator is a relativistic generalization of the quantum harmonic oscillator. In particular, the square of the Hamiltonian for the Dirac oscillator yields the Klein–Gordon equation with a potential of the form (ar 2+bL⋅S), where a and b are constants. To obtain the Dirac oscillator, a ‘‘minimal substitution’’ is made in the Dirac equation, where the ordinary derivative is replaced with a covariant derivative. However, a very unusual feature of the covariant derivative in this case is that the potential is a nontrivial element of the Clifford algebra. A theory which naturally gives rise to gauge potentials which are nontrivial elements of the Clifford algebra is that based on local automorphism invariance. An exact solution of the pure automorphism gauge field equations which reproduces both the potential term and the mass term of the Dirac oscillator is presented herein.

Mappings which Preserve Idempotents, Local Automorphisms, and Local Derivations

AbstractIt is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

FINITE-DIMENSIONALITY OF THE GROUP OF CR AUTOMORPHISMS OF A STANDARD CR MANIFOLD, AND PROPER HOLOMORPHIC MAPPINGS OF SIEGEL DOMAINS

It is shown that local CR automorphisms of nondegenerate standard CR manifolds are rational and form a finite-dimensional Lie group. It is established that proper holomorphic mappings of nondegenerate Siegel domains are biholomorphic and rational. Bibliography: 14 titles.

Moding of superconformal algebras

Different modings of an algebra correspond to different boundary conditions on the fields occuring in the operator product expansions. Some of these modings can be transformed into one another by local automorphisms. This is applied to an N=4 superconformal algebra with an SO(4) Kač-Moody and a parameter γ. For a general value of γ there is only one independent algebra, for γ= 1 2 there are two of them. It is also shown how this algebra contains known N=1, N=2 and N=3 superconformal algebras as subalgebras.

LOCAL AUTOMORPHISMS AND MAPPINGS OF SMOOTH STRICTLY PSEUDOCONVEX HYPERSURFACES

Suppose given a smooth strictly pseudoconvex hypersurface not equivalent to the sphere, a point on the surface, and a neighborhood of the point. It is shown that all local automorphisms of the surface defined in an arbitrary neighborhood of a fixed point can be linearized by a suitable choice of local coordinates. It is also shown that given two convergent sequences of real analytic nonspherical hypersurfaces it is possible to extract a convergent subsequence from any sequence of mappings between them. Bibliography: 20 titles.

ON THE LINEARIZATION OF AUTOMORPHISMS OF A REAL ANALYTIC HYPERSURFACE

It is proved that relative to certain normal coordinates any local automorphism of a nonspherical real-analytic hypersurface in Cn is a fractional-linear transformation. For a hypersurface in C3 a system of normal coordinates is constructed relative to which the entire stability group consists only of fractional-linear transformations. Bibliography: 13 titles.

Representation of inverse semigroups by local automorphisms and multi-automorphisms of groups and rings

Representation of inverse semigroups by local automorphisms and multi-automorphisms of groups and rings

LINEARIZABILITY OF AUTOMORPHISMS OF NON-SPHERICAL SURFACES

In this paper local automorphisms of real analytic hypersurfaces in complex spaces are studied. It is proved that for a strictly pseudoconvex hypersurface not biholomorphically equivalent to a sphere every local automorphism is a linear mapping in special coordinates. The equation of the surface has the Moser normal form in these coordinates. Bibliography: 8 titles.

Extension of KMS states and chemical potential

We present an algebraic description of the concept of chemical potential for a general compact gauge groupG, as a first step in the classification of thermodynamical equilibrium states of a given temperature. Adopting first the usual setting of a field algebra ℱ containing the observable algebra\(\mathfrak{A}\) as its gauge invariant part, we establish the following results (i) the existence and uniqueness, up to gauge, of τ-weakly clustering states ф of ℱ extending a given such state ω of\(\mathfrak{A}\) (τ an asymptotically abelian automorphism group of ℱ commuting withG) (ii) in the case of an ω faithful and β-KMS for a time evolution commuting with τ, and of a time-invariant ф, the fact that ф is β-KMS for a one-parameter group of time and gauge whose gauge part lies in the center of the stabilizerGφ of ф. (iii) a description of the general case where ф is neither time invariant nor faithful: ф is then in general vacuum-like in directions of the gauge space governed by an “asymmetry subgroup”. We further analyze the representations and von Neumann algebras determined by ω, ф and the gauge average\(\bar \omega \) of ф. The covariant representation generated by\(\bar \omega \) is shown to be obtained by inducing up fromGφ toG the representation generated by ф. Finally we present, for the case whereG is ann-dimensional torus, an intrinsic description of the chemical potential in terms of cocycle Radon-Nicodym derivatives of the state ω w.r.t. its (quasi equivalent) transforms by localized automorphisms of\(\mathfrak{A}\). Our main result (ii) is established using two independant techniques, the first making systematic use of clustering properties, the second relying on the analysis of representations. Both proofs are basically concerned with Tannaka duality—the second with a version thereof formulated in Robert's theory of Hilbert spaces in a von Neumann algebra.

Local Automorphisms are Differential Operators on Some Banach Spaces

If $E$ belongs to a certain category of Banach spaces (the ${B^\infty }$-smooth spaces) which include Hilbert spaces and if $F$ is any normed space, it is proved that any local linear automorphism of ${C^\infty }(E,F)$ is a differential operator. This generalizes a result of J. Peetre when $E = {R^n}$ and $F = R$.

Local automorphisms are differential operators on some Banach spaces

If E E belongs to a certain category of Banach spaces (the B ∞ {B^\infty } -smooth spaces) which include Hilbert spaces and if F F is any normed space, it is proved that any local linear automorphism of C ∞ ( E , F ) {C^\infty }(E,F) is a differential operator. This generalizes a result of J. Peetre when E = R n E = {R^n} and F = R F = R .

Fields, observables and gauge transformations II

We wish to study the construction of charge-carrying fields given the representation of the observable algebra in the sector of states of zero charge. It is shown that the set of those covariant sectors which can be obtained from the vacuum sector by acting with “localized automorphisms” has the structure of a discrete Abelian group . An algebra of fields $$\mathfrak{F}$$ can be defined on the Hilbert space of a representation π of the observable algebra $$\mathfrak{A}$$ which contains each of the above sectors exactly once. The dual group of acts as a gauge group on $$\mathfrak{F}$$ in such a way that $$\pi (\mathfrak{A})$$ is the gauge invariant part of $$\mathfrak{F}.\mathfrak{F}$$ is made up of Bose and Fermi fields and is determined uniquely by the commutation relations between spacelike separated fields.

Fields, observables and gauge transformations I

Starting from an algebra of fields $$\mathfrak{F}$$ and a compact gauge group of the first kind ℊ, the observable algebra $$\mathfrak{A}$$ is defined as the gauge invariant part of $$\mathfrak{F}$$ . A gauge group of the first kind is shown to be automatically compact if the scattering states are complete and the mass and spin multiplets have finite multiplicity. Under reasonable assumptions about the structure of $$\mathfrak{F}$$ it is shown that the inequivalent irreducible representations of $$\mathfrak{A}$$ (“sectors”) which occur are in one-to-one correspondence with the inequivalent irreducible representations of ℊ and that all of them are “strongly locally equivalent”. An irreducible representation of $$\mathfrak{A}$$ satisfies the duality property only if the sector corresponds to a 1-dimensional representation of ℊ. If ℊ is Abelian the sectors are connected to each other by localized automorphisms.