Let $\alpha$ and $\beta$ be two Furstenberg transformations on $2$-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.

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.

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 produce a complete description of the lattice of gauge-invariant ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph $\Lambda$. We provide a condition on $\Lambda$ under which every ideal is gauge-invariant. We give conditions on $\Lambda$ under which $C^*(\Lambda)$ satisfies the hypotheses of the Kirchberg--Phillips classification theorem.

For monotone complete $C^*$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed $C^*$-subalgebra, the relation $X = AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX + XA \subset X$ and $XX^* + X^*X \subset A$ and a set of certain partial isometries $s$ in the ``normalizer" of $A$ in $B$, and similarly for the map $s \mapsto \Ad s$ between the latter set and a set of certain ``partial $*$-automorphisms" of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type ${\rm II}_1$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the $C^*$-algebra version of these results.

The decomposability number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in $\m$. In this paper, we explore the close connection between $\dec(\m)$ and the cardinal level of the Mazur property for the predual $\m_*$ of $\m$, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group $G$ as the group algebra $\lone$, the Fourier algebra $A(G)$, the measure algebra $M(G)$, the algebra $\luc^*$, etc. We show that for any of these von Neumann algebras, say $\m$, the cardinal number $\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$ are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of $G$: the compact covering number $\kg$ of $G$ and the least cardinality $\bg$ of an open basis at the identity of $G$. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra $\ag^{**}$.

We show that certain $C^*$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed-product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that if $G$ is a hyperbolic group and if $\ell$ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a ``compact quantum metric space''). We show that a convenient framework is that of filtered $C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^*$-algebras.

We prove a symmetric imprimitivity theorem for commuting proper actions of locally compact groups $H$ and $K$ on a $C^*$-algebra.

Let $A$ be an amenable separable $C^*$-algebra and $B$ be a non-unital but $\sigma$-unital simple $C^*$-algebra with continuous scale. We show that two essential extensions $\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [\tau_1]=[\tau_2] \text{ in } KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B)$. Using $KL(A, M(B)/B)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories.

We are concerned with a unital separable nuclear purely infinite simple $C^{*}$-algebra\ $A$ satisfying UCT with a Rohlin flow, as a continuation of~\cite{Kismh}. Our first result (which is independent of the Rohlin flow) is to characterize when two {\em central} projections in $A$ are equivalent by a {\em central} partial isometry. Our second result shows that the K-theory of the central sequence algebra $A'\cap A^\omega$ (for an $\omega\in \beta\N\setminus\N$) and its {\em fixed point} algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in~ \cite{Kismh}.

Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$, respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in $\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be bounded families. We show the following equality $$ \lim_{i,\U} \lim_{j,\U'} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U} \Big\| \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\|_{L_p(\M_i\otimes \N_j)} . $$ For $p=1$ this Fubini type result is related to the local reflexivity of duals of $C^*$-algebras. This fails for $p=\infty$.

We study the K-homology of the rotation algebras $A_{\theta}$ using the six-term cyclic sequence for the K-homology of a crossed product by ${\bf Z}$. In the case that $\theta$ is irrational, we use Pimsner and Voiculescu's work on AF-embeddings of the $A_{\theta}$ to search for the missing generator of the even K-homology.

We study the type decomposition and the rectangular AFD property for $W^*$-TRO's. Like von Neumann algebras, every $W^*$-TRO can be uniquely decomposed into the direct sum of $W^*$-TRO's of type $I$, type $II$, and type $III$. We may further consider $W^*$-TRO's of type $I_{m, n}$ with cardinal numbers $m$ and $n$, and consider $W^*$-TRO's of type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$. It is shown that every separable stable $W^*$-TRO (which includes type $I_{\infty,\infty}$, type $II_{\infty, \infty}$ and type $III$) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for $W^*$-TRO's. One of our major results is to show that a separable $W^*$-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular

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~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann 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 $1\leq p\leq 2$.

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.

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A.~Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras.

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.

We investigate representations of the Cuntz algebra $\mathcal{O}_2$ on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi_{P_1}$. We pay corresponding to these representations and the Fock special attention to the vector states on $\mathcal{O}_2$ vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra $\mathcal{F}_2$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi_{P_1}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to $\mathcal{F}_2$. The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated with these representations of $\mathcal{O}_2$ are also considered.

It is shown that simple stably projectionless $\C^S*$-algebras which are inductive limits of certain specified building blocks with trivial $\K$-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form $$ \{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\}, $$ where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$. Furthermore we prove existence and uniqueness theorems for $*$-homomorphisms between such algebras and we identify the range of the invariant.

We prove a number of results about the stable and particularly the real ranks of tensor products of \ca s under the assumption that one of the factors is commutative. In particular, we prove the following: {\raggedright \begin{enumerate}[(5)] \item[(1)] If $X$ is any locally compact $\sm$-compact Hausdorff space and $A$ is any \ca, then\break $\RR \bigl( C_0 (X) \otimes A \bigr) \leq \dim (X) + \RR(A)$. \item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$. \item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$ for any unital \ca\ $A$. \item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) = 1$, and $K_1 (A) = 0$, then\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \item[(5)] There is a simple separable unital nuclear \ca\ $A$ such that $\RR(A) = 1$ and\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \end{enumerate}}

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras.

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.