Gelfand-Naimark Theorem of Commutative Hopf C^＊ -Algebras

Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.

LOCAL AUTOMORPHISMS OF SEMISIMPLE ALGEBRAS AND GROUP ALGEBRAS

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.

C^＊-Structure of Quantum Double for Finite Hopf C^＊-Algebra

Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.

Elementary C^(*)-Algebras and Haagerup Tensor Products

In this note,we give a new characterization of elementary C^(*)-algebras in terms of completely compact maps and Haagerup tensor products.

Asymptotic Direct System of C^(*)-algebras and Locally AF-algebra

We define the direct limit of the asymptotic direct system of C^(*)-algebras and give some properties of it.Finally,we prove that a C^(*)-algebra is a locally AF-algebra,if and only if it is the direct limit of an asymptotic direct system of finite-dimensional C^(*)-algebras.

Simple generalized inductive limits of C^(*)-algebras

We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self absorbing C^(*)-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_(2)),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.

A geometric characterisation of real C^(*)-algebras

We characterise the positive cone of a real C^(*)-algebra geometrically.Given an open coneΩin a real Banach space V,with the closureΩ,we show thatΩis the interior of the positive cone of a unital real C^(*)-algebra if and only if it is a Finsler symmetric cone with an orientable extension,which is equivalent to the condition that V is,in an equivalent norm,the Hermitian part of a unital real C^(*)-algebra with the positive coneΩ.

Positive combinations of projections in von Neumann algebras and purely infinite simple C*-algebras

We give an overview of the question： which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann Mgebras, on purely infinite simple C^＊-algebras, and on their associated multiplier algebras.

Classification of Homomorphisms from C(X)to Simple C~*-Algebras of Real Rank Zero

Let A be a unital simple C-algebra of real rank zero,stable rank one,with weakly unperforated K0（A）and unique normalized quasi-trace τ,and let X be a compact metric space.We show that two monomorphisms Φ,Ψ:C（X）→A are approximately unitarily equivalent if and only if Φ and Ψ induce the same element in KL（C（X）,A）and the two linear functionals τ ο Φ and τ ο Φ are equal.We also show that,with an injectivity condition,an almost multiplicative morphism from C（X） into A with vanishing KK-obstacle is close to a homomorphism.

Certain Cuntz Semigroup Properties of Certain Crossed Product C^*-algebras

We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.

THE K-THEORY AND THE CLASSIFICATION OF THE IRRATIONAL ROTATION GROUPOID C^(*)-ALGEBRAS

The authors determine the invariants of the irrational rotationC^(*)-algebras over theL-shaped domain in C^(2) by the maximal radical series.

Some Results on Inner Quasidiagonal C^*-algebras

In the current article,we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again.We also show that a just-infinite C^*-algebra is quasidiagonal if and only if it is inner quasidiagonal.Finally,we compute the topological free entropy dimension in just-infinite C^*-algebras.

On C^(*)-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).

INHERITANCE OF DIVISIBILITY FORMS A LARGE SUBALGEBRA

Let A be an infinite dimensional stably finite unital simple separable C^(*)-algebra.Let B■A be a stably(centrally)large subalgebra in A such that B is m-almost divisible(m-almost divisible,weakly(m,n)-divisible).Then A is 2(m+1)-almost divisible(weakly m-almost divisible,secondly weakly(m,n)-divisible).

The (＊＊)-Haagerup Property for C^＊-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki's question(2013), and then obtain several results of(**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the(**)-Haagerup property. Some heredity results concerning the(**)-Haagerup property are also proved.

Von Neumann Regularity and Quadratic Conorms in JB^＊-triples and C^＊-algebras

We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.

On the Decomposition Theorems for C^(*)-Algebras

Elliott dimension drop interval algebra is an important class among all C^*-algebras in the classification theory.Especially,they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth.In this paper,the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra.Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth.

A Perturbation of Jensen *-Derivations from K(H)into K(H)

Let＇s take H as an infinite-dimensional Hilbert space and K（H） be the set of all compact operators on H. Using Spectral theorem for compact self-adjoint operators, we prove the Hyers-Ulam stability of Jensen z-derivations from K（H） into K（H）.

On a geometric realization of C^＊-algebras

Further to the functional representations of C^＊-algebras proposed by R. Cirelli and A. Manik, we consider the uniform Kahler bundle （UKB） description of some C^＊-algebraic subjects. In particular, we obtain a one-to- one correspondence between closed ideals of a C^＊-algebra A and full uniform Kahler subbundles over open subsets of the base space of the UKB associated with A. In addition, we present a geometric description of the pure state space of hereditary C^＊-subalgebras and show that if B is a hereditary C^＊-subalgebra of A, the UKB of B is a kind of Kahler subbundle of the UKB of A. As a simple example, we consider hereditary C^＊-subalgebras of the C^＊-algebra of compact operators on a Hilbert space. Finally, we remark that each hereditary C^＊- subalgebra of A also can be naturally characterized as a uniform holomorphic Hilbert bundle.

A Note on the Monotone Product of Nuclear C^＊-Algebras

Given two nuclear C^＊-algebras A1 and A2 with states φ1 and φ2, we show that the monotone product C^＊-algebra A1 △→ A2 is still nuclear. Furthermore, if both the states φ1 and φ2 are faithful, then the monotone product ,A1 △→ A2 is nuclear if and only if the C^＊-algebras ,A1 and A2 both are nuclear.