Borut Zalar

Bicircular projections and characterization of Hilbert spaces

We prove that every JB* triple with rank one bicircular projection is a direct sum of two ideals, one of which is isometrically isomorphic to a Hilbert space.

On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra LP(G) of a compact metric group G is an isometric automorphism. In the last decade considerable work has been done concerning certain local maps of operator algebras. The originators of this research are Kadison and Larson. In [Kad], Kadison studied local derivations on a von Neumann algebra R. A continuous linear map on R7 is called a local derivation if it agrees with some derivation at each point (the derivations possibly differring from point to point) in the algebra. This investigation was motivated by the study of Hochschild cohomology of operator algebras. It was proved in [Kad] that in the above setting, every local derivation is a derivation. Independently, Larson and Sourour proved in [LaSo] that the same conclusion holds true for local derivations of the full operator algebra (3(X), where X is a Banach space. For other results on local derivations of various algebras see, for example, [Bre, BrSel, Cri, Shu, ZhXi]. Besides derivations, there is at least one additional very important class of transformations on Banach algebras which certainly deserves attention. This is the group of automorphisms. In [Lar, Some concluding remarks (5), p. 298], from the view-point of reflexivity, Larson raised the problem of local automorphisms (the definition should be self-explanatory) of Banach algebras. In his joint paper with Sourour [LaSo], it was proved that if X is an infinite dimensional Banach space, then every surjective local automorphism of B(X) is an automorphism (see also [BrSel]). For a separable infinite dimensional Hilbert space H, it was shown in [BrSe2] that the above conclusion holds true without the assumption on surjectivity, i.e. every local automorphism of 3B(JC) is an automorphism. For other results on local automorphisms of various operator algebras, we refer to [BaMo, Mol2, Mol3, Mol4]. Received by the editors March 4, 1998. 1991 Mathematics Subject Classification. Primary 43A15, 43A22, 46H99.

Reflexivity of the group of surjective isometries on some Banach spaces

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p -classes of compact operators. The ideas developed there can be used in l p -spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

Jordan *-derivation pairs and quadratic functionals on modules over *-rings

We give a new simple proof of Semrl’s recent representation theorem for quasi-quadratic functions acting on unital modules and then show that our approach also gives a certain extension of Semrl’s result.

A CLASS OF NONASSOCIATIVE ALGEBRAS ARISING FROM QUADRATIC ODEs

Abstract We present an algebraic part of our research on quadratic differential equations in 3-dimensional space, which possess a plane of critical points. Our goal is to provide a classification, up to isomorphism, of those algebras that arise from the above-mentioned systems via Markus construction.

Continuity of derivations on Mal'cev H*-algebras

A long time ago the concept of H *-algebra was introduced by Ambrose in [1] where the structure of complex associative H *-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [ 6 ]), Jordan algebras (in [ 5, 13, 14 ]), non-commutative Jordan algebras (in [ 5 ]), Lie algebras (in [ 3, 9, 10 ]) and Malcev algebras (in [ 2 ]).

Jordan ∗-derivations and quadratic functionals on octonion algebras

Let A be a ∗-algebra. An additive mapping E : A → A is called a Jordan ∗-derivation if E(X2) = E(x)x∗+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan ∗-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan ∗-derivation on the octonion algebra is of the form E(x)=ax∗-xa. In the terminology of earlier papers this means that every Jordan ∗-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan ∗-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan ∗-der...

NORM OF SUMS OF PEIRCE PROJECTIONS

Peirce projections P1, P1/2 and P0 are one of the fundamental technical tools in the theory of JB*-triples. It is well known that all three of them are contractive. We show that the sum of two Peirce projections need not be contractive. We also give the upper estimate for the norm of such a sum, valid in all JC*-triples, and present a conjecture of the exact value of this norm, based on some numerical experiments.