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1. CMB 2010 (vol 54 pp. 141)

Kim, Sang Og; Park, Choonkil
 Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if $\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and $\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra. Keywords:preservers, Jordan automorphisms, invertible operators, zero productsCategories:47B48, 47A10, 46H10

2. CMB 2002 (vol 45 pp. 309)

Xia, Jingbo
 Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$. Categories:46H10, 47B35, 47C05

3. CMB 2001 (vol 44 pp. 504)

Zhang, Yong
 Weak Amenability of a Class of Banach Algebras We show that, if a Banach algebra $\A$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\A$ implies the ($2m+1$)-weak amenability of $\A$ for all $m\geq 1$. Keywords:$n$-weak amenability, left ideals, left bounded approximate identityCategories:46H20, 46H10, 46H25