We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces.

We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $\|P\| < 2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.

Let $A$ be the inductive limit of a system $$A_{1}\xrightarrow{\phi_{1,2}}A_{2} \xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd $$ with $A_n = \bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will prove that $A$ can be written as another inductive limit $$B_1\xrightarrow{\psi_{1,2}} B_2 \xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd $$ with $B_n = \bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The result is trivial if one allows the spaces $Y_{n,i}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras (see \cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are graphs is due to the third named author \cite{Li1}.

Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\mathcal{A}\longrightarrow E$, the composition $j \circ T$ factors through a diagonal operator from $l^{2}$ into $l^{1}$. In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2
Keywords:$C^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym property
Categories:46L50, 47D15

A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are ``chaotic.'' While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of $C^*$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario.

We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin.

We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.

For a locally compact group $G$, the convolution product on the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra $\nN(L^p(G))$ and relate them to some properties of the group $G$, such as compactness, finiteness, discreteness, and amenability.

We show that Poincar\'e inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0
Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential norm
Categories:47B38, 47B10, 46E40, 46E15

It is known that any non-archimedean Fr\'echet space of countable type is isomorphic to a subspace of $c_0^{\mathbb{N}}$. In this paper we prove that there exists a non-archimedean Fr\'echet space $U$ with a basis $(u_n)$ such that any basis $(x_n)$ in a non-archimedean Fr\'echet space $X$ is equivalent to a subbasis $(u_{k_n})$ of $(u_n)$. Then any non-archimedean Fr\'echet space with a basis is isomorphic to a complemented subspace of $U$. In contrast to this, we show that a non-archimedean Fr\'echet space $X$ with a basis $(x_n)$ is isomorphic to a complemented subspace of $c_0^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of the following spaces: $c_0$, $c_0 \times \mathbb{K}^{\mathbb{N}}$, $\mathbb{K}^{\mathbb{N}}$, $c_0^{\mathbb{N}}$. Finally, we prove that there is no nuclear non-archimedean Fr\'echet space $H$ with a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear non-archimedean Fr\'echet space $Y$ is equivalent to a subbasis $(h_{k_n})$ of $(h_n)$.

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg: A uniform algebra equipped with its canonical, {\it i.e.}, minimal, operator space structure is operator amenable if and only if it is a commutative $C^\ast$-algebra.

We show that if $\mathcal{A}$ is a class of $C^\ast$-algebras for which the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$ is weakly stable for the class $\mathcal{B}$ that contains $\mathcal{A}$ and all the inductive limits that can be constructed with the $C^\ast$-algebras in $\mathcal{A}$. A set of formal relations $\mathcal{R}$ is said to be {\it weakly stable\/} for a class $\mathcal{C}$ of $C^\ast$-algebras if, in any $C^\ast$-algebra $A\in \mathcal{C}$, close to an approximate representation of the set $\mathcal{R}$ in $A$ there is an exact representation of $\mathcal{R}$ in $A$.

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an automorphism of $G_i$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in \Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is $\gamma_i$. As a consequence, we prove that every $\mathbb{Z}_2$-graded countable module over the representation ring $R (\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant $K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on a unital Kirchberg algebra~$A$. Along the way, we prove that every not necessarily finitely generated $\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and $\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts either trivially or by multiplication by $-1$.

We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fr\'echet differentiable. We give a general result on the Fr\'echet differentiability of $f\circ T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces of Lipschitz functions.

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.

In the sequel we construct simple, unital, separable, stable, amenable $C^{*}$-algebras for which the ordered $K_{0}$-group is strongly perforated and group isomorphic to $Z$. The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised.

We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple $C^*$-algebras. The first result of this kind was shown by Powers for the UHF algbras some 30 years ago.

Let $A_\theta$ be the universal $C^*$-algebra generated by two unitaries $U$, $V$ satisfying $VU=e^{2\pi i\theta} UV$ and let $\Phi$ be the antiautomorphism of $A_\theta$ interchanging $U$ and $V$. The $K$-theory of $R_\theta=\{a\in A_\theta:\Phi(a)=a^*\}$ is computed. When $\theta$ is irrational, an inductive limit of algebras of the form $M_q(C(\mathbb{T})) \oplus M_{q'} (\mathbb{R}) \oplus M_q(\mathbb{R})$ is constructed which has complexification $A_\theta$ and the same $K$-theory as $R_\theta$.

We determine when an automorphism of a subfactor of type III$_0$ with finite index is non-strongly free in the sense of C.~Winsl\o w in terms of the modular endomorphisms introduced by M.~Izumi.

We study the dimension of ``random'' Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from \cite{LMS}, to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much ``weaker'' randomness of ``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also add some relative information on ``phase transition''.

We exhibit examples of $F$-spaces with trivial dual which are isomorphic to its quotient by a line, thus solving a problem in Rolewicz's {\it Metric Linear Spaces}.

An $\AF$ flow is a one-parameter automorphism group of an $\AF$ $C^*$-algebra $A$ such that there exists an increasing sequence of invariant finite dimensional sub-$C^*$-algebras whose union is dense in $A$. In this paper, a classification of $C^*$-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/path-space construction, and one in terms of a modified $K_0$ functor.