The relative commutant of separable C-algebras of real rank zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C^∗-algebra in its ultrapower depend on the choice of the ultrafilter?     

Isomorphism of Hilbert modules over stably finite C-algebras

It is shown that if A is a stably finite C^∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C^∗-algebra that are not isomorphic.     

The Banach-Lie algebra of multiplication operators on a JBW-triple

The Banach-Lie algebra L(A) of multiplication operators on the JB^∗-triple A is introduced and it is shown that the hermitian part Lh(A) of L(A) is a unital GM-space the base of the dual cone in the dual GL-space ^∗(Lh(A)) of which is affine isomorphic and weak^∗-homeomorphic to the state space of L(A). In the case in which A is a JBW^∗-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space Lh(A) satisfy

0?D(u,u)+D(v,v)?idA,     

Stochastic representation of partial differential inclusions

Partial differential inclusions of the form and ψ∈LFGu+cu, where LFG is a uniformly parabolic second order set-valued operator, are considered. In particular, basing on diffusions properties of weak solutions to stochastic differential inclusions, some existence and representation theorems for solutions of such type partial differential inclusions are given.     

Weak exactness for dual operator spaces

We introduce a new tensor product and study the weak^∗ condition C^′, which is also called weak^∗ exactness, for dual operator spaces. Our definition of weak^∗ condition C^′ is equivalent to Kirchberg's notion of weak exactness in the case of von Neumann algebras. We also study the connection of weak^∗ exact W^∗-TROs with their linking von Neumann algebras and study the structure of exact (respectively, nuclear) W^∗-TROs.     

Nearness approximation space based on axiomatic fuzzy sets

The approximation space model was originally proposed by Pawlak (1982) [19]. It was Or?owska who first observed that approximation spaces serves as a formal counterpart of perception, or observation [16, §2, p. 8], in which approximations provide a means of approximating one set of objects with another set of objects using the indiscernibility relation. Topology has been used to enrich the original model of an approximation space as well as more recent models of generalized approximation spaces. In this paper, an extension of th e topology neighborhood based on AFS (Axiomatic Fuzzy Sets) theory is introduced, and some interesting properties are given. Furthermore, a new generalized approximation space model is established with two application examples, which can be used to deal with information tables with many category features and viewed as a multi-granulations form of nearness approximation space models.     

Isomorphisms between C-ternary algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in C^∗-ternary algebras and of derivations on C^∗-ternary algebras for the following Cauchy-Jensen additive mappings:

(0.1)     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

On weakly central C-algebras

A unital C^∗-algebra A is weakly central if and only if for every x∈A there exists a sequence of elementary unital completely positive maps αn on A such that the sequence (αn(x)) converges to a central element.     

Positive functionals on BP*-algebras

A new class of locally convex algebras, called BP^*-algebras, is introduced. It is shown that this class properly includes MQ^*-algebras which were introduced and studied by the first author andR. Rigelhof [10]. Among other results, it is proved that each positive functional on a BP^*-algebraA is admissible but not necessarily continuous as shown by an example. However, ifA, in addition, is either (i) a Q-algebra, or (ii) has an identity and is barrelled, or (iii)A is endowed with the inductive limit topology, then each positive functional onA is continuous.This work was supported by an N.R.C. Grant.     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Large Fredholm triples

We give several equivalent condition for Busby extensions of a given algebra to be absorbing, considerably improving our earlier results [G.A. Elliott, D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001) 385-409], and establish sufficient conditions for Fredholm triples to be absorbing in a suitable sense. As an application of one of our criteria, we prove a multivariable Brown-Douglas-Fillmore type theorem.     

Theory of Almost Algebraic Poincaré Complexes and Local Combinatorial Hirzebruch Formula

We develop a theory of almost algebraic Poincaré complexes to write an analog of the Hirzebruch formula with nonflat coefficients for combinatorial manifolds.     

C n -almost automorphic solutions for partial neutral functional differential equations

In this work, we study the existence of C ⁿ -almost periodic solutions and C ⁿ -almost automorphic solutions (n?≥?1), for partial neutral functional differential equations. We prove that the existence of a bounded integral solution on ?⁺ implies the existence of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions.