We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous.

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.

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}.

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 $H^{\ev}$.)

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.

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.

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.