The action on the trace space induced by a generic automorphism of a suitable finite classifiable ${\mathrm {C}^*}$-algebra is shown to be chaotic and weakly mixing. Model ${\mathrm {C}^*}$-algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe $KK$-contractible stably projectionless ${\mathrm {C}^*}$-algebras as crossed products.

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C*-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C*-algebras. On the other hand, we show every finite-dimensional C*-algebra is computably categorical.

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$-algebras associated to them, in terms of the pair of original graphs.

What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing $\mathrm{C}^*$-algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm {B}({\mathrm {L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm {L}}^2(X)$, where X is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh’s ${\mathrm {H}}^\infty $ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to ${\mathrm {BMO}}$-spaces. We also give an answer to a question of Steen, Todorov, and Turowska on completely positive continuous Schur multipliers.

We investigate the notion of tracial ${\mathcal {Z}}$-stability beyond unital $\mathrm {C}^*$-algebras, and we prove that this notion is equivalent to ${\mathcal {Z}}$-stability in the class of separable simple nuclear $\mathrm {C}^*$-algebras.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Let P be a closed convex cone in $\mathbb{R}^d$ which is assumed to be spanning $\mathbb{R}^d$ and contains no line. In this article, we consider a family of CAR flows over P and study the decomposability of the associated product systems. We establish a necessary and sufficient condition for CAR flow to be decomposable. As a consequence, we show that there are uncountable many CAR flows which are cocycle conjugate to the corresponding CCR flows.

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$, introduced by Brzeziński and Szymański as graph $C^*$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$-algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

We prove the first orbit equivalence superrigidity results for actions of type III$_\unicode{x3bb} $ when $\unicode{x3bb} \neq 1$. These actions arise as skew products of actions of dense subgroups of $\operatorname {SL}(n,\mathbb {R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.

For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz–Pimsner $C^*$-algebra $\mathcal {O}_X$ has nuclear dimension $1$ when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X, which also recovers an exact sequence, discovered before by Carlsen and Eilers.

It is shown that the colored isomorphism class of a unital, simple, $\mathcal {Z}$-stable, separable amenable C$^*$-algebra satisfying the universal coefficient theorem is determined by its tracial simplex.

We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal {D}}_E$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then, we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal {O}}_n$ that are not inner conjugate to the diagonal ${\mathcal {D}}_n$.

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen in 1988. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.

For amenable discrete groupoids $\mathcal {G}$ and row-finite directed graphs E, let $(\mathcal {G},E)$ be a self-similar groupoid and let $C^*(\mathcal {G}, E)$ be the associated $C^*$-algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$. Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.

Given an irreducible lattice $\Gamma $ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma $-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal {L}(\Gamma )$, and for the $\Gamma $-invariant unital $C^*$-subalgebras of the reduced group $C^*$-algebra $C^*_{\mathrm {red}}(\Gamma )$. We use these results to show that: (i) every $\Gamma $-invariant von Neumann subalgebra of $\mathcal {L}(\Gamma )$ is generated by a normal subgroup; and (ii) given a weakly mixing unitary representation $\pi $ of $\Gamma $, every $\Gamma $-equivariant conditional expectation on $C^*_\pi (\Gamma )$ is the canonical conditional expectation onto the $C^*$-subalgebra generated by a normal subgroup.

Let G be a locally compact unimodular group, and let $\phi $ be some function of n variables on G. To such a $\phi $, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded $(p_1,\ldots ,p_n)$-norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map $L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb {R}, S_{p_2}) \rightarrow L_{1}(\mathbb {R}, S_{1})$, whenever $p_1$ and $p_2$ are such that $\frac {1}{p_1} + \frac {1}{p_2} = 1$. A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.

We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^\star $-algebra, and if $\phi :A\to \mathbb C $ is a continuous function satisfying $ \phi (x)\phi (y) \in \sigma (xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi $ is a character of A or $-\phi $ is a character of A.

We provide an abstract characterization for the Cuntz semigroup of unital commutative AI-algebras, as well as a characterization for abstract Cuntz semigroups of the form $\operatorname {\mathrm {Lsc}} (X,\overline {\mathbb {N}})$ for some $T_1$-space X. In our investigations, we also uncover new properties that the Cuntz semigroup of all AI-algebras satisfies.