For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$ (containing the rationals) it is shown that the group $K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where $A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) = U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense $G_\delta$ set of parameters $\theta$. The main computational tool in this paper is a group homomorphism $\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of $H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$ is fully determined for each $\theta$. (We conjecture that this subspace is all of $H^{\ev}$.)

We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite II$_1$ factor $R$ in [E1]---which arise by considering unitary braid representations with certain properties---to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples---where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A$, $B$, $C$, $D$. We also compute the (first) relative commutants.

The concept of $R$-diagonal element was introduced in \cite{NS2}, and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to $R$-diagonality, which relies on freeness with amalgamation. The class of $R$-diagonal elements is enlarged to contain examples living in non-tracial $*$-probability spaces, such as the generalized circular elements of \cite{Sh1}.

Giordano, Putnam and Skau showed that the transformation group $C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra, and classified it by its $K$-theory. For approximately inner automorphisms that preserve $C(X)$, we will determine their classes in the Ext and OrderExt groups, and introduce a new invariant for the closure of the topological full group. We will also prove that every automorphism in the kernel of the homomorphism into the Ext group is homotopic to an inner automorphism, which extends Kishimoto's result.

We use some results about stable relations to show that some of the simple, stable, projectionless crossed products of $O_2$ by $\bR$ considered by Kishimoto and Kumjian are inductive limits of type I $C^*$-algebras. The type I $C^*$-algebras that arise are pullbacks of finite direct sums of matrix algebras over the continuous functions on the unit interval by finite dimensional $C^*$-algebras.

We prove that pre-classifiable (see 3.1) simple nuclear tracially AF \CA s (TAF) are classified by their $K$-theory. As a consequence all simple, locally AH and TAF \CA s are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple TAF \CA s with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and $K_0 (A)_+$ is locally finitely generated, then $A \otimes Q \cong B \otimes Q$, where $Q$ is the UHF-algebra with the rational $K_0$. Classification results (with restriction on $K_0$-theory) for the above \CA s are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple TAF \CA s with the unique normalized trace satisfying the UCT and with $K_1(A) = K_1(B)$, and $A$ and $B$ have the same rational (scaled ordered) $K_0$, then $A \cong B$. Similar results are also obtained for some cases in which $K_0$ is non-divisible such as $K_0(A) = \mathbf{Z} [1/2]$.

A simple $\C^*$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition---induced by addition of orthogonal projections---form the additive semi-group $$ \{0,2,3,\dots\}. $$ (This is a particularly simple instance of the phenomenon of perforation of the ordered $\K_0$-group, which has long been known in the commutative case---for instance, in the case of the four-sphere---and was recently observed by the second author in the case of a simple $\C^*$-algebra.)

We study estimates of the type $$ \Vert \phi(D) - \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D - D_0 \Vert^{\alpha}, \quad \alpha = \frac12, 1 $$ where $\phi(t) = t(1 + t^2)^{-1/2}$, $D_0 = D_0^*$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $\calM$, $D - D_0$ is a bounded self-adjoint linear operator from $\calM$ and $(1 + D_0^2)^{-1/2} \in \emt$, where $\emt$ is a symmetric operator space associated with $\calM$. In particular, we prove that $\phi(D) - \phi(D_0)$ belongs to the non-commutative $L_p$-space for some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the non-commutative weak $L_r$-space for some $r \in [1,p)$. In the case $\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this result continues to hold under the weaker assumption $(1 + D_0^2)^{-1/2} \in \calC_p$. This may be regarded as an odd counterpart of A.~Connes' result for the case of even Fredholm modules.

Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where $\theta$ is a fixed real number. Let $\sigma$ denote the Fourier automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$, and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote the associated $C^\ast$-crossed product. It is shown that there is a canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each $\theta$ given by nine canonical modules. The unbounded trace functionals of $B_\theta$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when $\theta$ is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense $G_\delta$ set of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is generated by the nine classes constructed here.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a $C^*$-coaction $\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a discrete group $G$ to a $C^*$-coaction $\Ind\delta\colon\Ind D\to \Ind D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over $C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

An {\it odd unbounded\/} (respectively, $p$-{\it summable}) {\it Fredholm module\/} for a unital Banach $\ast$-algebra, $A$, is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded self-adjoint operator on $H$ satisfying: \item{(1)} $(1+D^2)^{-1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{-(p/2)}\bigr) <\infty$), and \item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense $\ast-$subalgebra of $A$. If $u$ is a unitary in the dense $\ast-$subalgebra mentioned in (2) then $$ uDu^\ast=D+u[D,u^{\ast}]=D+B $$ where $B$ is a bounded self-adjoint operator. The path $$ D_t^u:=(1-t) D+tuDu^\ast=D+tB $$ is a ``continuous'' path of unbounded self-adjoint ``Fredholm'' operators. More precisely, we show that $$ F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{-{1\over 2}} $$ is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The {\it spectral flow\/} of this path $\{F_t^u\}$ (or $\{ D_t^u\}$) is roughly speaking the net number of eigenvalues that pass through $0$ in the positive direction as $t$ runs from $0$ to $1$. This integer, $$ \sf(\{D_t^u\}):=\sf(\{F_t^u\}), $$ recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory class [$u$]. We use I.~M.~Singer's idea (as did E.~Getzler in the $\theta$-summable case) to consider the operator $B$ as a parameter in the Banach manifold, $B_{\sa}(H)$, so that spectral flow can be exhibited as the integral of a closed $1$-form on this manifold. Now, for $B$ in our manifold, any $X\in T_B(B_{\sa}(H))$ is given by an $X$ in $B_{\sa}(H)$ as the derivative at $B$ along the curve $t\mapsto B+tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer: $$ \alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr) $$ is a closed $1$-form. For any piecewise smooth path $\{D_t=D+B_t\}$ with $D_0$ and $D_1$ unitarily equivalent we show that $$ \sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}} (D_t)(1+D_t^2)^{-m}\Bigr)\,dt $$ the integral of the $1$-form $\alpha$. If $D_0$ and $D_1$ are not unitarily equivalent, we must add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form: $$ \sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt $$ for $n\geq{{p-1}\over 2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D(1+D^2 )^{-{1\over 2}}$. We prove simultaneously a type II version of our results.

Examples of simple, separable, unital, purely infinite $C^\ast$-algebras are constructed, including: \item{(1)} some that are not approximately divisible; \item{(2)} those that arise as crossed products of any of a certain class of $C^\ast$-algebras by any of a certain class of non-unital endomorphisms; \item{(3)} those that arise as reduced free products of pairs of $C^\ast$-algebras with respect to any from a certain class of states.

In this paper we consider the following problem: Given a unital \cs\ $A$ and a collection of elements $S$ in the identity component of the invertible group of $A$, denoted \ino, characterize the group of finite products of elements of $S$. The particular $C^*$-algebras studied in this paper are either unital purely infinite simple or of the form \tenp, where $A$ is any \cs\ and $K$ is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents ($1+$ nilpotent), positive invertibles and symmetries ($s^2=1$). First we determine the groups of finite products for each collection of elements in \tenp. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for \tenp, is that for $A$ unital purely infinite and simple, \ino\ is generated by each of these collections of elements.

Let $A$ be a simple $C^*$-algebra with real rank zero, stable rank one and weakly unperforated $K_0(A)$ of countable rank. We show that a monomorphism $\phi\colon C(S^2) \to A$ can be approximated pointwise by homomorphisms from $C(S^2)$ into $A$ with finite dimensional range if and only if certain index vanishes. In particular, we show that every homomorphism $\phi$ from $C(S^2)$ into a UHF-algebra can be approximated pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra with finite dimensional range. As an application, we show that if $A$ is a simple $C^*$-algebra of real rank zero and is an inductive limit of matrices over $C(S^2)$ then $A$ is an AF-algebra. Similar results for tori are also obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$ for lower dimensional spaces is also studied.

We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit.