Bernard Russo

Solution of the contractive projection problem

In this paper we show that the class of J*-algebras (a class of concrete Jordan triple systems) is stable under the action of norm one projections. This result constitutes a general solution to a problem considered by the authors and others. Specifically our main result is: THEOREM 2. Let P be an arbitrary contractive projection defined on a J*-algebra M. Then P(M) is a Jordan triple system in the triple product (a, b, c) -+ (a, b, c) = P(f(ab*c t cb*a)), for a, b, c E P(M); and (P(M), ( )) has a faithful representation as a J”-algebra.

Derivations on Real and Complex JB*-Triples

These three questions had all been answered in the binary cases. Question 1 was answered affirmatively by Sakai [17] for C∗-algebras and by Upmeier [23] for JB -algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW -algebras. Question 3 was answered by Upmeier [23] for JB -algebras, and it follows trivially from the Kadison–Sakai answer to question 2 in the case of C∗-algebras. In the ternary case, both question 1 and question 3 were answered by Barton and Friedman in [3] for complex JB ∗-triples. In this paper, we consider question 2 for real and complex JBW ∗-triples and question 1 and question 3 for real JB ∗-triples. A real or complex JB ∗-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite-dimensional JB ∗-triple has the inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2 and 3 determine which of the infinite-dimensional real or complex Cartan factors have the inner derivation property.

The norm of the ^{}-Fourier transform on unimodular groups

We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real Une as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest. Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian.

Affine structure of facially symmetric spaces

In [ 7 ], the authors proposed the problem of giving a geometric characterization of those Banach spaces which admit an algebraic structure. Motivated by the geometry imposed by measuring processes on the set of observables of a quantum mechanical system, they introduced the category of facially symmetric spaces . A discrete spectral theorem for an arbitrary element in the dual of a reflexive facially symmetric space was obtained by using the basic notions of orthogonality, protective unit, norm exposed face, symmetric face, generalized tripotent and generalized Peirce projection , which were introduced and developed in this purely geometric setting.

Hankel operators in the Dixmier class

Abstract We study holomorphic functions / in the unit ball for which the small Hankel operator hf belongs to the Dixmier class.

Contractive projections and operator spaces

We define a family of Hilbertian operator spaces Hnk, 1≤k≤n, containing the row and column Hilbert spaces Rn, Cn and show that an atomic subspace X⊂B(H) which is the range of a contractive projection on B(H) is isometrically completely contractive to an l∞-sum of the Hnk and Cartan factors of types 1 to 4. We also give a classification up to complete isometry of w∗-closed atomic JW∗-triples which have no infinite-dimensional rank 1 w∗-closed ideal.

On compactness of composition operators in Hardy spaces of several variables

Characterizations of compactness are given for holomorphic com- position operators on Hardy spaces of a strongly pseudoconvex domain.

Boundedness of completely additive measures with application to 2-local triple derivations

We prove a Jordan version of Dorofeevs boundedness theorem for completely additive measues and use it to show that every (not necessarily linear nor continuous) 2-local triple derivation on a continuous JBW*-triple is a triple derivation.

The norm of the Lp-Fourier transform III compact extensions

Abstract A theorem of Hausdorff Young type is proved for integral operators in the setting of gage spaces. This theorem is used to show that the norm of the L p -Fourier transform on unimodular groups is stable under compact extension.

Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains

SummaryWe obtain trace ideal criteria for 0<p<∞ for holomorphic composition operators acting on the weighted Bergman spacesAα2(Ω) of a Bounded symmetric diomain Ω in ℂn.