2,386 research outputs found
Order isomorphisms between cones of JB-algebras
In this paper we completely describe the order isomorphisms between cones of
atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an
algebraic direct summand of the so-called engaged and disengaged part. On the
cone of the engaged part every order isomorphism is linear and the disengaged
part consists only of copies of . Furthermore, in the setting of
general JB-algebras we prove the following. If either algebra does not contain
an ideal of codimension one, then every order isomorphism between their cones
is linear if and only if it extends to a homeomorphism, between the cones of
the atomic part of their biduals, for a suitable weak topology
A Common Framework for Restriction Semigroups and Regular *-Semigroups
Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sided versions of the restriction property. From a varietal perspective, these classes of semigroups–more generally, the corresponding classes of Ehresmann semigroups–derive from reducts of inverse semigroups, now taking a to a+=aa−1 (or, dually, to a∗=a−1a, or in the two-sided version, to both). In this paper the notion of restriction semigroup is generalized to P-restriction semigroup, derived instead from reducts of regular ∗-semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] P-Ehresmann semigroups. The first main theorem is an abstract characterization of the posets P of projections of each type of such semigroup as ‘projection algebras’. The second main theorem, at least in the two-sided case, is that for every P-restriction semigroup S there is a P-separating representation into a regular ∗-semigroup, namely the ‘Munn’ semigroup on its projection algebra, consisting of the isomorphisms between the algebra’s principal ideals under a modified composition. This theorem specializes to known results for restriction semigroups and for regular ∗-semigroups. A consequence of this representation is that projection algebras also characterize the posets of projections of regular ∗-semigroups. By further characterizing the sets of projections ‘internally’, we connect our universal algebraic approach with the classical approach of the so-called ‘York school’. The representation theorem will be used in a sequel to show how the structure of the free members in some natural varieties of (P-)restriction semigroups may easily be deduced from the known structure of associated free inverse semigroups
Jordan weak amenability and orthogonal forms on JB*-algebras
We prove the existence of a linear isometric correspondence between the
Banach space of all symmetric orthogonal forms on a JB-algebra
and the Banach space of all purely Jordan generalized derivations
from into . We also establish the existence of a
similar linear isometric correspondence between the Banach spaces of all
anti-symmetric orthogonal forms on , and of all Lie Jordan
derivations from into
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