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1. CMB 2011 (vol 55 pp. 821)

Perez-Garcia, C.; Schikhof, W. H.
New Examples of Non-Archimedean Banach Spaces and Applications
The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.

Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
Categories:46S10, 12J25

2. CMB 2007 (vol 50 pp. 588)

Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John
Cohomological Dimension and Schreier's Formula in Galois Cohomology
Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$.

Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups
Categories:12G05, 12G10

3. CMB 2003 (vol 46 pp. 388)

Lin, Huaxin
Tracially Quasidiagonal Extensions
It is known that a unital simple $C^*$-algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital $C^*$-algebras with tracial topological rank zero that have real rank other than zero. Let $0\to J\to E\to A\to 0$ be a short exact sequence of $C^*$-algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a $C^*$-algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

Keywords:tracially quasidiagonal extensions, tracial rank
Categories:46L05, 46L80

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