The Ext--Group of Unitary Equivalence Classes of Unital Extensions

Let \A be a unital separable nuclear $C^*$--algebra which belongs to the bootstrap category $\N$ and \B be a separable stable $C^*$--algebra. In this paper, we consider the group \Ext_u(\A,\B) consisting of the unitary equivalence classes of unital extensions \tau\colon\A\rightarrow Q(\B). The relation between \Ext_u(\A,\B) and \Ext(\A,\B) is established. Using this relation, we show the half--exactness of \Ext_u(\cdot,\B) and the (UCT) for \Ext_u(\A,\B). Furthermore, under certain conditions, we obtain the half--exactness and Bott periodicity of \Ext_u(\A,\cdot).Comment: 16 pages Acta Math. Sinica (English series)(to appear

Relatively spectral homomorphisms and K-injectivity

Let \A and \B be unital Banach algebras and \phi\colon\A\to\B be a unital continuous homomorphism. We prove that if $\phi$ is relatively spectral (i.e., there is a dense subalgebra $X$ of \A such that \sp_\B(\phi(a))=\sp_\A(a) for every $a\in X$) and has dense range, then $\phi$ induces monomorphisms from K_i(\A) to K_i(\B), $i=0,1$.Comment: 6 page

Approximate diagonalization of self--adjoint matrices over C(M)C(M)

Let $M$ be a compact Hausdorff space. We prove that in this paper, every self--adjoint matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$ and \HO^2(M,\mathbb Z)\cong 0. Using this result, we show that every unitary matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$, \HO^1(M,\mathbb Z)\cong\HO^2(M,\mathbb Z)\cong 0 when $M$ is a compact metric space.Comment: 11 page

Perturbation analysis of bounded homogeneous generalized inverses on Banach spaces

Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field.Comment: 14 page

Perturbation analysis of AT,S(2)A_{T,S}^{(2)} on Banach spaces

In this paper, the perturbation problems of $A_{T,S}^{(2)}$ are considered. By virtue of the gap between subspaces, we derive the conditions that make the perturbation of $A_{T,S}^{(2)}$ is stable when $T,S$ and $A$ have suitable perturbations. At the same time, the explicit formulas for perturbation of $A_{T,S}^{(2)}$ and new results on perturbation bounds are obtained.Comment: 13 pages, Electronic Journal of Linear Algebra (accepted

Perturbation analysis of AT,S(2)A_{T,S}^{(2)} on Hilbert spaces

In this paper, we investigate the perturbation analysis of $A_{T,S}^{(2)}$ when $T,\,S$ and $A$ have some small perturbations on Hilbert spaces. We present the conditions that make the perturbation of $A_{T,S}^{(2)}$ is stable. The explicit representation for the perturbation of $A_{T,S}^{(2)}$ and the perturbation bounds are also obtained.Comment: 10 page

The characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras

In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators.Comment: 17 page

Zonal polynomials and hypergeometric functions of quaternion matrix argument

We define zonal polynomials of quaternion matrix argument and deduce some important formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix $W\sim\mathbb{Q}W(n,\Sigma)$, respectively.Comment: 22 pages. Communications in Statistics - Theory and Methods (appear

The expression of Moore--Penrose inverse of A−XY∗A-XY^*

Let $K,\,H$ be Hilbert spaces and let $L(K,H)$ denote the set of all bounded linear operators from $K$ to $H$. Let $A \in L(H)\triangleq L(H,H)$ with $R(A)$ closed and $X,Y \in L(K,H)$ with $R(X)\subseteq R(A),R(Y)\subseteq R(A^*)$. In this short note, we give some new expressions of the Moore--Penrose inverse $(A-XY^*)^+$ of $A-XY^*$ under certain suitable conditions.Comment: 6 page

Perturbation analysis for the generalized inverses with prescribed idempotents in Banach algebras

In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations.Comment: 18 page