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1. CJM 2012 (vol 65 pp. 559)
Extreme Version of Projectivity for Normed Modules Over Sequence Algebras We define and study the so-called extreme version of the notion of a
projective normed module. The relevant definition takes into account
the exact value of the norm of the module in question, in contrast
with the standard known definition that is formulated in terms of norm
topology.
After the discussion of the case where our normed algebra $A$ is just
$\mathbb{C}$, we concentrate on the case of the next degree of complication,
where $A$ is a sequence algebra, satisfying some natural conditions.
The main results give a full characterization of extremely projective
objects within the subcategory of the category of non-degenerate
normed $A$--modules, consisting of the so-called homogeneous modules.
We consider two cases, `non-complete' and `complete', and the
respective answers turn out to be essentially different.
In particular, all Banach non-degenerate homogeneous modules,
consisting of sequences, are extremely projective within the category
of Banach non-degenerate homogeneous modules. However, neither of
them, provided it is infinite-dimensional, is extremely projective
within the category of all normed non-degenerate homogeneous modules.
On the other hand, submodules of these modules, consisting of finite
sequences, are extremely projective within the latter category.
Keywords:extremely projective module, sequence algebra, homogeneous module Category:46H25 |
2. CJM 2010 (vol 62 pp. 845)
Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.
Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebra Categories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07 |
3. CJM 2007 (vol 59 pp. 966)
Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finite-dimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$-multiplier
norm.
Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 |