We show that certain purely infinite simple $\mathrm{C}^*$-algebras, including the Cuntz algebra $O_{\infty}$, are semiprojective. Some related results and conjectures are discussed, and some crossed product examples constructed.

We give a detailed survey of the theory of quasidiagonal $C^*$-algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about tensor products, quotients, extensions, free products, etc. of quasidiagonal $C^*$-algebras. We also point out how quasidiagonality is connected to some important open problems.

Let $0 \to I \to E \to B \to 0$ be a short exact sequence of $C^*$-algebras where $E$ is separable, $I$ is quasidiagonal (QD) and $B$ is nuclear, QD and satisfies the UCT. It is shown that if the boundary map $\partial : K_1(B) \to K_0(I)$ vanishes then $E$ must be QD also.

A Hahn-Banach type property for $K_0$ of QD $C^*$-algebras is also formulated. It is shown that every nuclear QD $C^*$-algebra has this $K_0$-Hahn-Banach property if and only if the boundary map $\partial : K_1(B) \to K_0(I)$ (from above) always completely determines when $E$ is QD in the nuclear case.

For any row finite directed graph $E$ there exists a universal $C^*$-algebra $C^*(E)$ ([KPR, KPRR]) generated by projections and partial isometries satisfying the Cuntz-Krieger $E$-relations. This class of graph algebras includes the Cuntz-Krieger algebras and all AF algebras up to stable isomorphisms([D]). In this paper we give conditions for $E$ under which the algebra $C^*(E)$ has stable rank one or real rank zero. A simple graph $C^*$-algebra is either AF or purely infinite, hence it is always extremally rich. We discuss the extremal richness of some graph $C^*$-algebras and present several examples of prime ones with finitely many closed ideals.

We review recent developments concerning the classification theory of aperiodic automorphisms of certain classes of simple $C^*$-algebras.

All $C^*$-algebras of sections of locally trivial $C^*$-algebra bundles over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}$ with fibres $M_c(A_{\omega})$ are constructed under the assumption that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras. It is shown that each $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}$ with fibres $M_c(A_{\omega})$ is stably isomorphic to $C(\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}) \otimes M_c(A_{\omega})$.

Let $A_{cd}$ be a $cd$-homogeneous $C^*$-algebra over $\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1} \times \mathbb{T}^{r+2}$ of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torus $\mathbb{S}_{\rho}^{cd}$ is defined by twisting $C^* (\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2})$ in $A_{cd} \otimes C^* (\mathbb{Z}^{m-2})$ by a totally skew multiplier $\rho$ on $\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2}$. We prove that $\mathbb{S}_{\rho}^{cd} \otimes M_{p\infty}$ is isomorphic to $C(\prod_{i=1}^{e} S^{2n_i} \times \prod_{j=1}^{s} S^{2k_j-1}) \otimes C^* (\widehat{\mathbb{T}^{r+2}} \times \mathbb{Z}^{m-2}, \rho) \otimes M_{cd} (\mathbb{C}) \otimes M_{p\infty}$ if and only if the set of prime factors of $cd$ is a subset of the set of those of $p$.

We give a survey of known and a few new results on stable $C^*$-algebras. Characterizations of stable $C^*$-algebras are described, it is decided for a number of operations on $C^*$-algebras whether or not they leave the class of stable $C^*$-algebras invariant, and the relation between this topic and the structure of simple $C^*$-algebras is discussed.

We review various properties of braiding in subfactor theory and their connection to nets of factors on $S^1$ particularly.

We will review a notion of Morita equivalence between subfactors, which is a variation of Morita equivalence in ring and module theory. The main result is stated as follows: for arbitrary two Morita equivalent subfactors of hyperfinite $\mathrm{II}_1$ factors with finite Jones index and finite depth we can always choose a finite dimensional nondegenerate commuting square which gives rise to the subfactors isomorphic to the original ones. As an application of Morita equivalence between subfactors in connection with recent developments of theory of finite dimensional weak $C^*$-Hopf algebras, we will make a brief comment about the 3-dimensional topological quantum field theories obtained from subfactors with finite index and finite depth.

We study relative positions of four subspaces in a Hilbert space. Gelfand-Ponomarev gave a complete classification of indecomposable systems of four subspaces in a finite-dimensional space. In this note we show that there exist uncountably many indecomposable systems of four subspaces in an infinite-dimesional Hilbert space. We extend a numerical invariant, called defect, for a certain class of systems of four subspaces using Fredholm index. We show that the set of possible values of the defect is $\{\frac{n}{3};\ n \in \mathbf{Z}\}$.