312 research outputs found
Positive definite hermitian mappings associated to tripotent elements
We give a simple proof of a meaningful result established by Y. Friedman and
B. Russo in 1985, whose proof was originally based on strong holomorphic
results. We provide a simple proof which is directly deduced from the axioms of
JB*-triples with techniques of Functional Analysis
A note on 2-local representations of C-algebras
We survey the results on linear local and 2-local homomorphisms and zero
products preserving operators between C-algebras, and we incorporate some
new precise observations and results to prove that every bounded linear 2-local
homomorphism between C-algebras is a homomorphism. Consequently, every
linear 2-local -homomorphism between C-algebras is a -homomorphism
On the unit sphere of positive operators
Given a C-algebra , let denote the set of those positive
elements in the unit sphere of . Let , and be
complex Hilbert spaces, where and are infinite-dimensional and
separable. In this note we prove a variant of Tingley's problem by showing that
every surjective isometry or
(respectively, ) admits a unique extension
to a surjective complex linear isometry from onto
(respectively, from onto ). This provides a positive answer to
a conjecture posed by G. Nagy [\emph{Publ. Math. Debrecen}, 2018]
Weak Banach-Saks property and Komlos theorem for preduals of JBW-triples
We show that the predual of a JBW-triple has the weak Banach-Saks
property, that is, reflexive subspaces of a JBW-triple predual are
super-reflexive. We also prove that JBW-triple preduals satisfy the
Koml\'os property (which can be considered an abstract version of the weak law
of large numbers). The results rely on two previous papers from which we infer
the fact that, like in the classical case of , a subspace of a
JBW-triple predual contains as soon as it contains uniform copies
of
Weak 2-local derivations on
We introduce the notion of weak-2-local derivation (respectively,
-derivation) on a C-algebra as a (non-necessarily linear) map
satisfying that for every and there
exists a derivation (respectively, a -derivation) ,
depending on , and , such that and . We prove that
every weak-2-local -derivation on is a linear derivation. We also
show that the same conclusion remains true for weak-2-local -derivations on
finite dimensional C-algebras
Linear isometries between real JB*-triples and C*-algebras
Let be a (not necessarily surjective) linear isometry between two
real JB-triples. Then for each there exists a tripotent in
the bidual, of such that \begin{enumerate}[] \item
, for all in
the real JB-subtriple, generated by ; \item The mapping
is a linear isometry.
\end{enumerate} Furthermore, when is a real C-algebra, the projection
satisfies that is an
isometric triple homomorphism. When and are real C-algebras and
is abelian of real type, then there exists a partial isometry such
that the mapping is an isometric triple
homomorphism. These results generalise, to the real setting, some previous
contributions due to C.-H. Chu and N.-C. Wong, and C.-H. Chu and M. Mackey in
2004 and 2005. We give an example of a non-surjective real linear isometry
which cannot be complexified to a complex isometry, showing that the results in
the real setting can not be derived by a mere complexification argument.Comment: to appear in Quart. J. Mat
Quasi-linear functionals determined by weak-2-local -derivations on
We prove that, for every separable complex Hilbert space , every
weak-2-local -derivation on is a linear -derivation. We also
establish that every (non-necessarily linear nor continuous) weak-2-local
derivation on a finite dimensional C-algebra is a linear derivation
On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions
Let be a compact Hausdorff space and let be a real or complex Hilbert
space with dim. We prove that the space of all
-valued continuous functions on , equipped with the supremum norm,
satisfies the Mazur--Ulam property, that is, if is any real Banach space,
every surjective isometry from the unit sphere of onto the
unit sphere of admits a unique extension to a surjective real linear
isometry from onto . Our strategy relies on the structure of
-module of and several results in JB-triple theory. For this
purpose we determine the facial structure of the closed unit ball of a real
JB-triple and its dual space
von Neumann algebra preduals satisfy the linear biholomorphic property
We prove that for every JBW-triple of rank , the symmetric part
of its predual reduces to zero. Consequently, the predual of every infinite
dimensional von Neumann algebra satisfies the linear biholomorphic
property, that is, the symmetric part of is zero. This solves a problem
posed by M. Neal and B. Russo in [Mathematica Scandinavica, to appear
The Mazur-Ulam property for commutative von Neumann algebras
Let be a -finite measure space. Given a Banach space
, let the symbol stand for the unit sphere of . We prove that the
space of all complex-valued measurable essentially
bounded functions equipped with the essential supremum norm, satisfies the
Mazur-Ulam property, that is, if is any complex Banach space, every
surjective isometry admits an
extension to a surjective real linear isometry . This conclusion is derived from a more general statement which assures that
every surjective isometry where is a Stonean
space, admits an extension to a surjective real linear isometry from
onto
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