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.

{Correspondence} between torsion-free connections with {nilpotent skew-symmetric curvature operator} and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

We give an explicit construction of polynomial (\emph{of arbitrary degree}) $(\alpha,\beta)$-metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all non-trivial projectively flat $(\alpha,\beta)$-metrics of constant flag curvature.

In this paper we study the Chen--Ruan cohomology ring of weighted projective spaces. Given a weighted projective space ${\bf P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3\nobreakdash-multi\-sector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen--Ruan cohomology ring of weighted projective space ${\bf P}^{5}_{1,2,2,3,3,3}$.

Let $A$ be a separable amenable purely infinite simple \CA which satisfies the Universal Coefficient Theorem. We prove that $A$ is weakly semiprojective if and only if $K_i(A)$ is a countable direct sum of finitely generated groups ($i=0,1$). Therefore, if $A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal G}\subset A$ satisfying the following: for any contractive positive linear map $L: A\to B$ (for any \CA $B$) with $ \|L(ab)-L(a)L(b)\|<\dt$ for $a, b\in {\mathcal G}$ there exists a homomorphism $h\from A\to B$ such that $ \|h(a)-L(a)\|<\ep$ for $a\in {\mathcal F}$.

We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the $C^\ast$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl( K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of $C_0(X)$, respectively. In this article we consider the possibility of these transitions for general $C^\ast$-algebras. The difficult part is to start with a pro-$C^\ast$-algebra $A$ and to construct a $C^\ast$-algebra $A_0$ such that $A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is possible are called {\it locally compact\/} and we have characterized them using a concept similar to that of an approximate identity.

In this paper, we investigate projectivity in the category of operator spaces. In particular, we show that the Fourier algebra of a locally compact group $G$ is operator biprojective if and only if $G$ is discrete.