Peirce gradings and Peirce inner ideals in JBW*-triple factors

A Peirce inner ideal J in an anisotropic Jordan*-triple A gives rise to a Peirce grading (J ₀, J ₁, J ₂) of A by defining

On range tripotents in JBW*-triples

Let B be a JBW*-triple, let A be a JB*-subtriple of B and let be the set of range tripotents relative to A. It is shown that, under certain conditions, the supremum of a family of range tripotents in coincides with that in the complete lattice of all tripotents in B. As a consequence, a sufficient condition for a tripotent to be a range tripotent relative to A is obtained. The action of isomorphisms on range tripotents is investigated, and an analysis of the suprema of families of spectral range tripotents leads to a generalization of a result known for open projections in W*-algebras. Received: 8 July 2008     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

It is shown that if P is a weak*-continuous projection on a JBW*-triple A with predual A _*, such that the range PA of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of PA, then P is contractive if and only if it commutes with all inner derivations of PA. This provides characterizations of 1-complemented elements in a large class of subspaces of A _* in terms of commutation relations.     

Faithful Inner Ideals in Jbw*-Triples

The kernel Ker(J) and the annihilator J^⊥ of a weak*-closed inner ideal J in a JBW*-triple A consist of the sets of elements a in A for which {J a J} and {J a A} are zero, respectively, and J is said to be faithful if, for every non-zero ideal I in A, I ∩ Ker (J) is non-zero. It is shown that every weak*-closed inner ideal J in A has a unique orthogonal decomposition into a faithful weak*-closed inner ideal f(J) and a weak*-closed ideal f (J)^⊥ ∩ J of A. The central structure of f ( J) is investigated and used to show that J has zero annihilator if and only if it coincides with the multiplier of f (J). The results are applied to the cases in which J is the Peirce-two or Peirce-zero space A₂(v) or A₀(v) corresponding to a tripotent v in A, and to the case in which the JBW*-triple A is a von Neumann algebra.     

<Emphasis Type="Italic">C</Emphasis>*-algebras generated by cancellative semigroups

A C*-algebra generated by a commuting family of isometries is a natural generalization of the Toeplitz algebra. We study the *-automorphisms and invariant ideals of the C*-algebra geerated by a semigroup.     

Special jordan H*-triple systems

This work, jointly with [9], completes the structure theory and classification of the Jordan H ^*- triple systems. The problem of describing the Jordan H ^*-triple systems is reduced in [5] to that of describing the topologically simple ones. Ruling out the finite-dimensional case, we have that any of these H ^*-triples has an underlying triple system structure of quadratic type (and these can be fully described), or it is the H ^*-triple system associated to the odd part of a topologically simple Z₂-graded Jordan H ^*-algebra, whose classification is given in [13].     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

The Facial and Inner Ideal Structure of a Real JBW*–Triple

Let B be a real JBW*–triple with predual B_* and canonical hermitification the JBW*–triple A It is shown that the set 𝒰(B)^∼ consisting of the partially ordered set 𝒰(B) of tripotents in B with a greatest element adjoined forms a sub–complete lattice of the complete lattice 𝒰(A)^∼ of tripotents in A with the same greatest element adjoined. The complete lattice 𝒰(B)^∼ is shown to be order isomorphic to the complete lattice ℱ_n(B_*1 of norm–closed faces of the unit ball B_*1 in B_* and anti–order isomorphic to the complete lattice ℱ_w*(B₁) of weak*–closed faces of the unit ball B₁ in B. Consequently, every proper norm–closed face of B_*1 is norm–exposed (by a tripotent) and has the property that it is also a norm–closed face of the closed unit ball in the predual of the hermitification of B. Furthermore, every weak*–closed face of B₁ is weak*–semi–exposed, and, if non–empty, of the form u + B₀(u)₁ where u is a tripotent in B and B₀(u)₁ is the closed unit ball in the zero Peirce space B₀(u) corresponding to u. A structural projection on B is a real linear projection R on B such that, for all elements a and b in B, {Ra b Ra}_B is equal to R{a Rb a}_B. A subspace J of B is said to be an inner ideal if {J B J}_B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker(J) defined to be the set of elements b in B such that, for all elements a in J, {a b a}_B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.     

Radon-Nikodým theorem for biweights and regular biweights on partial *-algebras

The first purpose of this paper is to investigate Radon-Nikodym theorem for biweights on partial *-algebra. Secondly, we study regularity of biweights on partial *-algebraA and show that a biweightϕ onA is decomposed intoϕ=ϕ _r+ϕ _s, whereϕ _r is a regular biweight onA andϕ _s is a singular biweight onA.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Collinear Systems and Normal Contractive Projections on JBW*-Triples

Given a family {x _k }_k∈K of elements x _k in the predual A _* of a JBW^*-triple A, such that the support tripotents e _k of x _k form a collinear system in the sense of [31], necessary and sufficient criteria for the existence of a contractive projection from A _*. onto the subspace are provided. Preparatory to these results, and interesting in itself, is a set of necessary and sufficient algebraic conditions upon a contractive projection P on A for its range PA to be a subtriple. The results also provide criteria for the range of a normal contractive projection on A to be a Hilbert space. Supported by the Irish Research Council for Science, Engineering and Technology, Grant No. R 9854.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals

We establish several generalisations of Urysohn's lemma in the setting of JB^∗-triples which provide full answers to Problems 1.12 and 1.13 in Fernández-Polo and Peralta (2007) [22]. These results extend the previous generalisations obtained by C.A. Akemann, G.K. Pedersen and L.G. Brown in the setting of C^∗-algebras. A generalised Kadison's transitivity theorem is established for finite sums of pairwise orthogonal compact tripotents in JBW^∗-triples. We introduce the notion of positively open tripotent in the bidual of a JB^∗-triple as an extension of a concept which was already considered in the setting of ternary rings of operators. We investigate the connections appearing between positively open tripotents and hereditary inner ideals.     

The arens closure of a nonassociative algebra

Given a nonassociative algebra A and an Arens pair A₁, A₂, for A, we identify a subalgcbra ? of A₂ with i (A) ? A ? A₂ and show that ? better reflects the algebraic structure ot A, in parti-cular. any multilinear identity satisfied by ? is also satisfied by ? Hence, ? is commutative or Lie when A is and Jordan when A is a Jordan algebra of characteristic not 2 or 3. Also, we list examples (1) where ? = End_D(V) for A a primitive, associative algebra with commuting ring D and irreducible faithful module V,(2) where ? is the norm closure of A in the arens algebra of all bounded functionals of the bounded functionals for a normed algebra A and (3) where ? is the Arens algebra of all bounded functionals of the bounded functionals with A again normed. Note that dif-ferent Arens closures can arise form the same choice of A, A₁, , A₂ since ? is determined by A, A₁, A₂ and subspaces A₃ ? A₂ ^*, A₄,?A₃ ^*.     

Limit algebras of differential forms in non-commutative geometry

Given a C*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω_∞ A and Ω_ε A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω_∞ A induced by a K-cycle on A are considered. The GNS-representation of Ω_∞ A defined by a d-dimensional non-commutative volume integral on a d ⁺-summable K-cycle on A is realized as the representation induced by the left action of A on Ω*A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).     

<Emphasis Type="Italic">C</Emphasis>*-algebras generated by semigroups

In this paper we study C*-algebras generated by a commuting family of isometric operators. Such algebras naturally generalize the Toeplitz algebra. We investigate *-automorphisms and ideals of C*-algebras generated by semigroups.     

Cycles of Spatial Discretizations of Shadowing Dynamical Systems

Let H(B,α) be the JBW^*-algebra of elements of a continuous W^*-algebra B invariant under the ^*-anti-automorphism α of B of order two. Then the mapping I → I ∩ H(B, α) is an order isomorphism from the complete lattice of α-invariant weak^* closed inner ideals in B onto the complete lattice of weak^* closed inner ideals in H(B, α), every one of which is of the form eH(B, α) α(e) for some unique projection e in B with α-invariant central support. A corollary of this result completely characterizes the weak^* closed inner ideals in any continuous JBW^*-triple.     

Partial Maps with Domain and Range: Extending Schein's Representation

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 *-derivations.