Unbounded C*-Seminorms and Biweights on Partial *-Algebras

Unbounded C*-seminorms generated by families of biweights on a partial *-algebra are considered and the admissibility of biweights is characterized in terms of unbounded C*-seminorms they generate. Furthermore, it is shown that, under suitable assumptions, when the family of biweights consists of all those ones which are relatively bounded with respect to a given C*-seminorm q, it can be obtained an expression for q analogous to that one which holds true for the norm of a C*-algebra.     

Manifolds of tripotents in JB*-triples

Abstract. We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball. Received July 7, 1998; in final form November 16, 1998     

*-representations on Banach *-algebras

We study notions of -bounded linear functionals and represent-
able functionals on Banach *-algebras. An equivalence between these two is established for general Banach *-algebras. In particular, we characterize -bounded linear functionals on Banach *-algebras with approximate identity and isometric involution. In addition, we prove a result on representation of -bounded positive linear functionals in terms of cyclic vectors for the corresponding *-representation.

     

Representations of Certain Banach C*-Modules

The possibility of extending the well known Gelfand–Naimark–Segal representation of *-algebras to certain Banach C*-modules is studied. For this aim the notion of modular biweight on a Banach C*-module is introduced. For the particular class of strict pre CQ*-algebras, two different types of representations are investigated.     

Spectral Properties of Partial *-Algebras

We continue our study of topological partial *algebras focusing our attention to some basic spectral properties. The special case of partial *-algebras of operators is examined first, in order to find sufficient hints for the study of the abstract case. The outcome consists in the selection of a class of topological partial *-algebras (partial GC*-algebras) that behave well from the spectral point of view and that allow, under certain conditions, a faithful realization as a partial O*-algebra.     

Pure states and P-commutative Banach *-algebras

A Banach *-algebra A with bounded approximate identity is shown to be P-commutative if the spectrum of each element x in A coincides with the set of values at x of all pure states of A. An isomorphism theorem for P-commutative Banach *-algebras is established, and a result on the computation of the norm of a positive functional on a symmetric, P-commutative, Banach *-algebra with bounded approximate identity with bound one is proved.     

A note on banach partial *-algebras

A Banach partial *-algebra is a locally convex partial *-algebra whose total space is a Banach space. A Banach partial *-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of such objects and display a number of examples, namely L ^p -like function spaces and spaces of operators on Hilbert scales.     

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.     

Linear Functional Equations in Banach Modules over a C *-Algebra

The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C ^*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C ^*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.     

On Two-sided Locally Convex H*-algebras

We introduce (left, right, two-sided) locally convex H*-algebras, and we give conditions under which an one-sided locally convex H*-algebra turns to be a two-sided one (actually, a locally convex H*-algebra). We also give an example of a proper right locally convex H*-algebra with a (right) involution, which is not a left involution and an example of a proper two-sided locally convex H*-algebra, which is not a locally convex H*-algebra. Moreover, we connect (via an Arens-Michael decomposition) a two-sided locally m-convex H*-algebra with the classical (Banach) two-sided H*-algebras. Further, we present conditions so that the left, right involutions be continuous, and we see when a twosided locally convex H*-algebra is a dual one. Finally, we present some properties of invariant ideals which play an important rôle in structure theory of two-sided locally convex H*-algebras.     

New Holomorphically Closed Subalgebras of C*-Algebras of Hyperbolic Groups

We present a new construction of dense, isospectral subalgebras of unconditional Banach algebras over word-hyperbolic groups. We study the algebras thus obtained and derive applications to delocalized L ²-invariants of closed Riemannian manifolds of negative curvature and to the local cyclic cohomology of the reduced group C*-algebras of word-hyperbolic groups.     

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.     

Equivalence of the Stone-Weierstrass conjectures for C* and JB*-algebras

We prove that the Stone-Weierstrass conjectures for C*-algebras and JB*-algebras are equivalent.     

Geometric characterizations of some classes of operators in C*-algebras and von Neumann algebras

We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.

     

Ring C*-algebras

We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples. Originally, our motivation comes from algebraic number theory.     

On Quasi-Chebyshevity Subsets of Unital Banach Algebras

In this paper,first,we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev.Examples of those algebras are given including the algebras of continuous functions on compact sets.We also see some results in C*-algebras and Hilbert C*-modules.Next,by considering some conditions,we study Chebyshev of subalgebras in C*-algebras.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

WQ*-algebras of measurable operators

Every C*-algebra $\mathfrak{A}$ has a faithful *-representation π in a Hilbert space $\mathcal{H}$ . Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.     

Active lattices determine AW*-algebras

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.