We consider a class $(A, S, \alpha)$ of dynamical systems, where $S$ is an Ore semigroup and $\alpha$ is an action such that each $\alpha_s$ is injective and extendible ({\it i.e.} it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group $G$ of $S$ constructed in \cite[Proposition~6.1]{LR2}. It turns out that the full crossed product by this coaction is isomorphic to $A\rtimes_\alpha S$. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

We show that the multi-sided inclusion $R^{\otimes l} \subset R$ of braid-type subfactors of the hyperfinite II$_1$ factor $R$, introduced in {\it Multi-sided braid type subfactors} [E3], contains a sequence of intermediate subfactors: $R^{\otimes l} \subset R^{\otimes l-1} \subset \cdots \subset R^{\otimes 2} \subset R$. That is, every $t$-sided subfactor is an intermediate subfactor for the inclusion $R^{\otimes l} \subset R$, for $2\leq t\leq l$. Moreover, we also show that if $t>m$ then $R^{\otimes t} \subset R^{\otimes m}$ is conjugate to $R^{\otimes t-m+1} \subset R$. Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.

The Cuntz-Krieger algebra $\mathcal{O}_B$ is defined for an arbitrary, possibly infinite and infinite valued, matrix $B$. A graph $C^{\ast}$-algebra $G^{\ast} (E)$ is introduced for an arbitrary directed graph $E$, and is shown to coincide with a previously defined graph algebra $C^{\ast} (E)$ if each source of $E$ emits only finitely many edges. Each graph algebra $G^{\ast} (E)$ is isomorphic to the Cuntz-Krieger algebra $\mathcal{O}_B$ where $B$ is the vertex matrix of~$E$.

The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$.

H.~O.~Kim has shown that contrary to the case of $H^p$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into $H^p$, $0 < p \leq \infty$.

We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov's question negatively.

A local version of $\VMO$ is defined, and the local Hardy space $\hone$ is shown to be its dual. An application to weak-$*$ convergence in $\hone$ is proved.

We show that, if a Banach algebra $\A$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\A$ implies the ($2m+1$)-weak amenability of $\A$ for all $m\geq 1$.

We examine Hilbert bimodules which possess a (generally unbounded) involution. Topics considered include a linking algebra representation, duality, locality, and the role of these bimodules in noncommutative differential geometry

Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities.

Irrational rotation $C^*$-algebras have an inductive limit decomposition in terms of matrix algebras over the space of continuous functions on the circle and this decomposition can be chosen to be invariant under the flip automorphism. It is shown that the flip is essentially the only toral automorphism with this property.

The singular spectrum of $u$ in a convolution equation $\mu * u = f$, where $\mu$ and $f$ are tempered ultradistributions of Beurling or Roumieau type is estimated by $$ SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f. $$ The same is done for $SS_{*}u$.

Let $\cZ$ be the unital simple nuclear infinite dimensional $C^*$-algebra which has the same Elliott invariant as $\bbC$, introduced in \cite{JS}. A $C^*$-algebra is called $\cZ$-stable if $A \cong A \otimes \cZ$. In this note we give some necessary conditions for a unital simple $C^*$-algebra to be $\cZ$-stable.

We consider \cst-algebras which are inductive limits of finite direct sums of copies of $ C([0,1]) \otimes \Otwo$. For such algebras, the lattice of closed two-sided ideals is proved to be a complete invariant.

We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the ``tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman).

We give new examples of weak Hilbert spaces.

If $A$ is a prime C$^*$-algebra, $a \in A$ and $\lambda$ is in the numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^* (a-\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of $W(a)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general C$^*$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the C$^*$-convex hull of $a$ are characterized.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant. As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet spaces $E$ and $F$ that we have the topological identity $$ \bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H} \Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega \biggr). $$ Analogous results for the compact open topology have long been established. We also give an example to show that the (Qno) hypothesis on both $E$ and $F$ is necessary.

The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type~I and Type~II spectral flow.

We construct two non-isomorphic nuclear, stably finite, real rank zero $C^\ast$-algebras $E$ and $E'$ for which there is an isomorphism of ordered groups $\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible with all the coefficient transformations. The $C^\ast$-algebras $E$ and $E'$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras with similar properties.

We define embedding of an $n$-dimensional normed space into $L_{-p}$, $0
Categories:42A82, 46B04, 46F12, 60E07

We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established.