1. CJM Online first
 De Bernardi, Carlo Alberto; Veselý, Libor

Tilings of normed spaces
By a tiling of a topological linear space $X$ we mean a
covering of $X$ by at least two closed convex sets,
called tiles, whose nonempty interiors are
pairwise disjoint.
Study of tilings of infinitedimensional spaces initiated in
the
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
spaces,
and then apply these results to study existence of tilings of
normed and Banach spaces by tiles possessing
certain smoothness or rotundity properties. For a Banach space
$X$,
our main results are the following.
1. $X$ admits no tiling by FrÃ©chet smooth bounded tiles.
2. If $X$ is locally uniformly rotund (LUR), it does not admit
any tiling by balls.
3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$
uncountable, do admit
a tiling by pairwise disjoint LUR bounded tiles.
Keywords:tiling of normed space, FrÃ©chet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$space Categories:46B20, 52A99, 46A45 

2. CJM 2013 (vol 66 pp. 1143)
 Plevnik, Lucijan; Šemrl, Peter

Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space
Let $\mathcal{H}$ and $\mathcal{K}$ be infinitedimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.
Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents Categories:46B20, 47B49 

3. CJM 2013 (vol 66 pp. 373)
 Kim, Sun Kwang; Lee, Han Ju

Uniform Convexity and BishopPhelpsBollobÃ¡s Property
A new characterization of the uniform convexity of
Banach space is obtained in the sense of BishopPhelpsBollobÃ¡s
theorem. It is also proved that the couple of Banach spaces $(X,Y)$
has the bishopphelpsbollobÃ¡s property for every banach space $y$
when $X$ is uniformly convex. As a corollary, we show that the
BishopPhelpsBollobÃ¡s theorem holds for bilinear forms on
$\ell_p\times \ell_q$ ($1\lt p, q\lt \infty$).
Keywords:BishopPhelpsBollobÃ¡s property, BishopPhelpsBollobÃ¡s theorem, norm attaining, uniformly convex Categories:46B20, 46B22 

4. CJM 2010 (vol 62 pp. 827)
 Ouyang, Caiheng; Xu, Quanhua

BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vectorvalued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$uniformly convex, where $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$uniformly smooth norm.
Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces Categories:46E40, 42B25, 46B20 

5. CJM 2010 (vol 62 pp. 595)
6. CJM 2009 (vol 61 pp. 124)
 Dijkstra, Jan J.; Mill, Jan van

Characterizing Complete Erd\H os Space
The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zerodimensional, nowhere zerodimensional, Polishable ideals, submeasures on $\omega$, $\R$trees, linefree groups in Banach spaces Categories:28C10, 46B20, 54F65 

7. CJM 2008 (vol 60 pp. 1108)
 LopezAbad, J.; Manoussakis, A.

A Classification of Tsirelson Type Spaces
We give a complete classification of mixed Tsirelson spaces
$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ for finitely many pairs of
given compact and hereditary families $\mathcal F_i$ of finite sets of
integers and $0<\theta_i<1$ in terms of the CantorBendixson indices
of the families $\mathcal F_i$, and $\theta_i$ ($1\le i\le r$). We
prove that there are unique countable ordinal $\alpha$ and
$0<\theta<1$ such that every block sequence of
$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$ has a subsequence equivalent to a
subsequence of the natural basis of the
$T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of
comparison in between two of these mixed Tsirelson spaces.
Categories:46B20, 05D10 

8. CJM 2007 (vol 59 pp. 1029)
 Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M.

The Geometry of $L_0$
Suppose that we have the unit Euclidean ball in
$\R^n$ and construct new bodies using three operations  linear
transformations, closure in the radial metric, and multiplicative
summation defined by $\x\_{K+_0L} = \sqrt{\x\_K\x\_L}.$ We prove
that in dimension $3$ this procedure gives all originsymmetric convex
bodies, while this is no longer true in dimensions $4$ and higher. We
introduce the concept of embedding of a normed space in $L_0$ that
naturally extends the corresponding properties of $L_p$spaces with
$p\ne0$, and show that the procedure described above gives exactly the
unit balls of subspaces of $L_0$ in every dimension. We provide
Fourier analytic and geometric characterizations of spaces embedding
in $L_0$, and prove several facts confirming the place of $L_0$ in the
scale of $L_p$spaces.
Categories:52A20, 52A21, 46B20 

9. CJM 2007 (vol 59 pp. 63)
 Ferenczi, Valentin; Galego, Elói Medina

Some Results on the SchroederBernstein Property for Separable Banach Spaces
We construct a continuum of mutually
nonisomorphic
separable Banach spaces which are complemented in each other.
Consequently, the SchroederBernstein Index of any of these spaces is
$2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and
B. Maurey in 1997.
We also use classical descriptive set theory methods, as in some
work of the first author and C. Rosendal, to improve some results
of P. G. Casazza and
of N. J. Kalton on the
SchroederBernstein Property for
spaces with an unconditional finitedimensional Schauder
decomposition.
Keywords:complemented subspaces,, SchroederBernstein property Categories:46B03, 46B20 

10. CJM 2006 (vol 58 pp. 820)
 Moreno, J. P.; Papini, P. L.; Phelps, R. R.

Diametrically Maximal and Constant Width Sets in Banach Spaces
We characterize diametrically maximal and constant width
sets in $C(K)$, where $K$ is any compact Hausdorff space. These
results are applied to prove that the sum of two diametrically
maximal sets needs not be diametrically maximal, thus solving a
question raised in a paper by Groemer. A~characterization of
diametrically maximal sets in $\ell_1^3$ is also given, providing
a negative answer to Groemer's problem in finite dimensional
spaces. We characterize constant width sets in $c_0(I)$, for
every $I$, and then we establish the connections between the Jung
constant of a Banach space and the existence of constant width
sets with empty interior. Porosity properties of families of sets
of constant width and rotundity properties of diametrically
maximal sets are also investigated. Finally, we present some
results concerning nonreflexive and Hilbert spaces.
Categories:52A05, 46B20 

11. CJM 2004 (vol 56 pp. 472)
 Fonf, Vladimir P.; Veselý, Libor

InfiniteDimensional Polyhedrality
This paper deals with generalizations of the notion of a polytope to infinite
dimensions. The most general definition is the following: a bounded closed
convex subset of a Banach space is called a \emph{polytope} if each of its
finitedimensional affine sections is a (standard) polytope.
We study the relationships between eight known definitions
of infinitedimensional
polyhedrality. We provide a complete isometric
classification of them, which gives
solutions to several open problems.
An almost complete isomorphic classification
is given as well (only one implication remains open).
Categories:46B20, 46B03, 46B04, 52B99 

12. CJM 2004 (vol 56 pp. 225)
 Blower, Gordon; Ransford, Thomas

Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals Categories:46B20, 46L52 

13. CJM 2000 (vol 52 pp. 999)
 Mankiewicz, Piotr

Compact Groups of Operators on Subproportional Quotients of $l^m_1$
It is proved that a ``typical'' $n$dimensional quotient $X_n$ of
$l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property
$$
\Average \int_G \Tx\_{X_n} \,dh_G(T) \geq
\frac{c}{\sqrt{n\log^3 n}} \biggl( n  \int_G \tr T \,dh_G (T)
\biggr),
$$
for every compact group $G$ of operators acting on $X_n$, where
$d_G(T)$ stands for the normalized Haar measure on $G$ and the average
is taken over all extreme points of the unit ball of $X_n$. Several
consequences of this estimate are presented.
Categories:46B20, 46B09 

14. CJM 1999 (vol 51 pp. 566)
 Ferenczi, V.

Quotient Hereditarily Indecomposable Banach Spaces
A Banach space $X$ is said to be {\it quotient hereditarily
indecomposable\/} if no infinite dimensional quotient of a subspace
of $X$ is decomposable. We provide an example of a quotient
hereditarily indecomposable space, namely the space $X_{\GM}$
constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we
provide an example of a reflexive hereditarily indecomposable space
$\hat{X}$ whose dual is not hereditarily indecomposable; so
$\hat{X}$ is not quotient hereditarily indecomposable. We also
show that every operator on $\hat{X}^*$ is a strictly singular
perturbation of an homothetic map.
Categories:46B20, 47B99 

15. CJM 1999 (vol 51 pp. 26)
 Fabian, Marián; Mordukhovich, Boris S.

Separable Reduction and Supporting Properties of FrÃ©chetLike Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chetlike
normals and $\epsilon$normals to arbitrary sets in general Banach
spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is
particularly helpful in Asplund spaces where every separable subspace
admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable
reduction method in Asplund spaces, we provide a new direct proof of a
nonconvex extension of the celebrated BishopPhelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$normals.
Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chetlike normals and subdifferentials, supporting properties, Asplund spaces Categories:49J52, 58C20, 46B20 

16. CJM 1997 (vol 49 pp. 1242)
 Randrianantoanina, Beata

$1$complemented subspaces of spaces with $1$unconditional bases
We prove that if $X$ is a complex strictly monotone sequence
space with $1$un\con\di\tion\al basis, $Y \subseteq X$ has no bands
isometric to $\ell_2^2$ and $Y$ is the range of normone projection from
$X$, then $Y$ is a closed linear span a family of mutually
disjoint vectors in $X$.
We completely characterize $1$complemented subspaces and normone
projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q <
\infty$.
Finally we give a full description of the subspaces that are spanned
by a family of disjointly supported vectors and which are
$1$complemented in (real or complex) Orlicz or Lorentz sequence
spaces. In particular if an Orlicz or
Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p <
\infty$ then the only subspaces
of $X$ which are $1$complemented and disjointly supported are the
closed linear spans of block bases with constant
coefficients.
Categories:46B20, 46B45, 41A65 
