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Lajos Molnar

Publications and source records attributed to Lajos Molnar.

36 records · Page 2Linked to original sources
arXiv2000

Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles

Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed.

math.FA↗
arXiv2000

*-semigroup endomorphisms of B(H)

Let H be a complex infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms $ϕ$ of B(H) which are uniformly continuous on every commutative C*-subalgebra. In particular, we obtain that if $ϕ$ satisfies $ϕ(0)=0$, then $ϕ$ is additive.

math.OA↗
arXiv2000

On some automorphisms of the set of effects on Hilbert space

The set of all efects on a Hilbert space has an affine structure (it is a convex set) as well as a multiplicative structure (it can be equipped with the so-called Jordan triple product). In this paper we describe the corresponding automorphisms of that set.

math.OA↗
arXiv2000

On isomorphisms of standard operator algebras

The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to Jordan triple isomorphisms of associative rings is automatically additive.

math.OA↗
arXiv1999

Multiplicative maps on ideals of operators which are local automorphisms

We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).

math.OA↗
arXiv1999

Some multiplicative preservers on $BH)$

In this paper we describe the form of those continuous multiplicative maps on B(H) (H being a separable complex Hilbert space of dimension not less than 3) which preserve the rank, or the corank. Furthermore, we characterize those continuous *-semigroup endomorphisms of B(H) which are spectrum non-increasing.

math.OA↗
arXiv1999

A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules

Let H be a Hilbert $C^*$-module over a matrix algebra A. It is proved that any function $T:H\to H$ which preserves the absolute value of the (generalized) inner product is of the form $Tf=ϕ(f)Uf$ $(f\in H)$, where $ϕ$ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner's classical unitary-antiunitary theorem for Hilbert modules.

math.FA↗
arXiv1998

Some linear preserver problems on B(H) concerning rank and corank

As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a result on the form of linear maps which preserve the left ideals in B(H).

math.OA↗
arXiv1998

An algebraic approach to Wigner's unitary-antiunitary theorem

We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C*-algebras.

math.OA↗
arXiv1997

Reflexivity of the automorphism and isometry groups of the suspension of $B(H)$

The aim of this paper is to show that the automorphism and isometry groups of the suspension of $B(H)$, $H$ being a separable infinite dimensional Hilbert space, are algebraically reflexive. This means that every local automorphism, respectively local surjective isometry of $C_0(\mathbb R)\otimes B(H)$ is an automorphism, respectively a surjective isometry.

math.FA↗
arXiv1997

Diameter preserving linear bijections of $C(X)$

The aim of this paper is to solve a linear preserver problem on the function algebra $C(X)$. We show that in case $X$ is a first countable compact Hausdorff space, every linear bijection $ϕ:C(X)\to C(X)$ having the property that $diam(ϕ(f)(X))=diam(f(X))$ $(f\in C(X))$ is of the form \[ ϕ(f)=τ\cdot f\circ φ+t(f)1 \qquad (f\in C(X)) \] where $τ$ is a complex number of modulus 1, $φ:X\to X$ is a homeomorphism and $t$ is a linear functional on $C(X)$.

math.FA↗
arXiv1997

Reflexivity of the automorphism and isometry groups of some standard operator algebras

In this paper we give an example of a proper standard C*-algebra (a proper C*-subalgebra of B(H) containing C(H)) whose automorphism and isometry groups are topologically reflexive. Furthermore, we prove that in the case of extensions of C(H) by separable commutative C*-algebras, these groups are algebraically reflexive. Concerning the most well-known extensions of C(H) by the algebra of all continuous complex valued functions on the perimeter of the unit disc, we show that the automorphism and isometry groups are topologically nonreflexive.

math.FA↗
arXiv1997

Linear maps on factors which preserve the extreme points of the unit ball

The aim of this paper is to characterize those linear maps from a von Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital *-homomorphism or a unital *-antihomomorphism.

math.FA↗