EP modular operators and their products

We study first EP modular operators on Hilbert C^?-modules and then we provide necessary and sufficient conditions for the product of two EP modular operators to be EP. These enable us to extend some results of Koliha (2000) [13] for an arbitrary C^?-algebra and the C^?-algebras of compact operators.     

A Galois correspondence for compact group actions on C-algebras

In this paper, we prove a Galois correspondence for compact group actions on C^?-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C^?-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ）× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

Groupoid structure of Toeplitz algebra over L-shaped domain

The structure of the groupoidG associated with the ToeplitzC ^*-algebraC ^*(Ω) of the L-shaped domain is discussed. The detailed characterization ofM _∞ by the classification of the closed subgroup of the Euclidean space is presented. Project supported partially by the National Natural Science Foundation of China, Fok Yingtung Educational Foundation and the Foundation of the State Education Commission of China.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

On Quantales and Spectra of C*-Algebras

We study properties of the quantale spectrum MaxA of an arbitrary unital C^*-algebra A. In particular we show that the spatialization of MaxA with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C^*-algebras.     

A bott periodicity map for crossed products of C*-algebras by discrete groups

A generalization is given of the canonical map from a discrete group into K ₁ of the group C ^*-algebra. Our map also generalizes Rieffel's construction of a projection in an irrational rotation C ^*-algebra.     

Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C ₀(ℝ²) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 788–796.Original Russian Text Copyright ©2005 by T. V. Shul’man.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

On the K-theory of the stable C-algebras from substitution tilings

We describe a method to compute the K-theory of the C^?-algebra arising from the stable equivalence relation in the Smale space associated to a substitution tiling, and give detailed computations for one- and two-dimensional examples. We prove that for one-dimensional tilings the group K₀ is always torsion free and give an example of a two-dimensional tiling such that K₀ has torsion.     

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.     

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.     

Geometric structure of dimension functions of certain continuous fields

For a continuous field of C^?-algebras A, we give a criterion to ensure that the stable rank of A is one. In the particular case of a trivial field this leads to a characterization of stable rank one, completing accomplishments by Nagisa, Osaka and Phillips. Further, for certain continuous fields of C^?-algebras, we study when the Cuntz semigroup satisfies the Riesz interpolation property, and we also analyze the structure of its functionals. As an application, we obtain a positive answer to a conjecture posed by Blackadar and Handelman in a variety of situations.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

Approximate Isomorphisms in <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C ^*-algebras: