Geometric unitaries in JB-algebras

In this note, we study geometric unitaries of JB-algebras (in particular, self-adjoint parts of C^*-algebras). By using order theoretical arguments, we show that the geometric unitaries of a JB-algebra are precisely the central algebraic unitaries (or central symmetries).     

Structure and representations of noncommutative C*-Jordan algebras

We show that a unital n.c. (noncommutative) JB^*-algebra has a faithful family of factor-representations of type I and determine the structure of n.c. JB^*-factors: A n.c. JB^*-factor is a commutative Jordan algebra, or flexible quadratic, or a quasi CC^*-algebra.     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

On normed Jordan algebras which are Banach dual spaces

Alfsen, Shultz, and Størmer have defined a class of normed Jordan algebras called JB-algebras, which are closely related to Jordan algebras of self-adjoint operators. We show that the enveloping algebra of a JB-algebra can be identified with its bidual. This is used to show that a JB-algebra is a dual space iff it is monotone complete and admits a separating set of normal states; in this case the predual is unique and consists of all normal linear functionals. Such JB-algebras (“JBW-algebras”) admit a unique decomposition into special and purely exceptional summands. The special part is isomorphic to a weakly closed Jordan algebra of self-adjoint operators. The purely exceptional part is isomorphic to C(X, M₃⁸) (the continuous functions from X into M₃⁸).     

Local Triple Derivations on C*-Algebras†

We prove that every bounded local triple derivation on a unital C*-algebra is a triple derivation. A similar statement is established in the category of unital JB*-algebras.     

The fixed point property in -triples and preduals of -triples

We prove that for every member X in the class of real or complex JB^*-triples or preduals of JBW^*-triples, the following assertions are equivalent:

(1) X has the fixed point property.

(2) X has the super fixed point property.

(3) X has normal structure.

(4) X has uniform normal structure.

(5) The Banach space of X is reflexive.

As a consequence, a real or complex C^*-algebra or the predual of a real or complex W^*-algebra having the fixed point property must be finite-dimensional.

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

On real forms of JB*-triples

We introduce real JB^*-triples as real forms of (complex) JB^*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB^*-triples is an isometry if and only if it commutes with the cube mappinga→a ³={aaa}. This generalizes a result of Dang for complex JB^*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB^*-triple numerical invariants that are respected by surjective linear isometries.     

Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a C*-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a C*-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.     

Bounded and Unitary Elements in Pro-C*-algebras

A pro-C*-algebra is a (projective) limit of C*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C*-algebras can be seen as non-commutative k-spaces. An element of a pro-C*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C*-algebra. In this paper, we investigate pro-C*-algebras from a categorical point of view. We study the functor (−) _b that assigns to a pro-C*-algebra the C*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (−) _b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand duality for commutative unital pro-C*-algebras is also presented.     

Topologies on BP-algebras

The class of locally convex ^*-algebra topologies on a BP^*-algebra which possess the same bounded hermitian idempotent subsets is considered and is shown to have a finest element. The cone of positive elements of a symmetric BP^*-algebra is studied and is seen to be closed in this finest topology. Order-bounded BP^*-algebras are considered, and it is seen that the positive linear functionals span the dual of such an algebra. The class of equivalent topologies on an order-bounded commutative BP^*-algebra for which every positive linear functional is continuous is considered, and there are found to be such topologies which are neither barreled nor Q-topologies, so the results of [6.], 15–28) are extended.     

Operator Jensen＇s Inequality on C＊-algebras

A real continuous function which is defined on an interval is said to beA-convex if it is convex on the set of self-adjoint elements,with spectra in the interval,in all matrix algebras of the unital C-algebra A.We give a general formation of Jensen’s inequality for A-convex functions.     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB?-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C?-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C?-algebra to a Banach bimodule is continuous.     

One-Parameter Groups Arising from Real Subspaces of Self-Dual Hilbert W*-Moduli

The purpose of this paper is to generalize the results of M. A. Rieffel and A. van Daele [8, §§ 1, 2, 3] for Hilbert W^*-moduli over commutative W^*-Algebras. Some special real subspaces of such Hilbert W^*-moduli and the related operators are investigated. Particularly, the relation is established between ^*weakly continuous unitary one-parameter groups of operators arising from them and the generalized K.M.S. condition. All key definitions are formulated without any commutativity supposition for the underlying W^*-algebra. The interpretation of these results is given for sets of continuous sections of “self-dual” locally trivial Hilbert bundles over hyperstonian compact spaces. At the end of this paper some aspects of the general noncommutative case are discussed.     

Spectral Doubling of Normal Operators and Connections with Antiunitary Operators

Equations such as AB = B ^T A have been studied in the finite dimensional setting in (Linear Algebra Appl 369:279–294, 2003). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB*J for some conjugation operator J in place of B ^T . Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B*K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators B for which AB = (JB*J)A and BA = A(JB*J) and also those B with KB = B*K for certain antiunitary operators K.