The subject of this article is operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from [Ans07, BL09, BL11, LS08] and extend a number of results from these articles to our more general setting. The results include the quadratic relation satisfied by the kernel of the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.

We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.

We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$, $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$. Our proof relies on Mei–Ricard’s results [MR16] regarding $\operatorname {L}^p$-boundedness (for all $1 < p < +\infty $) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan–Ioana–Kunnawalkam Elayavalli’s recent construction [CIKE22] to provide the first example of a $\mathrm {II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialised case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers.

Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the ${C^*}$-algebra of the skew product and a certain native coaction on the ${C^*}$-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the ${C^*}$-algebra of the original quiver.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

When $\mathcal {D}$ is strongly self-absorbing, we say an inclusion $B \subseteq A$ of C*-algebras is $\mathcal {D}$-stable if it is isomorphic to the inclusion $B \otimes \mathcal {D} \subseteq A \otimes \mathcal {D}$. We give ultrapower characterizations and show that if a unital inclusion is $\mathcal {D}$-stable, then $\mathcal {D}$-stability can be exhibited for countably many intermediate C*-algebras concurrently. We show that such unital embeddings between unital $\mathcal {D}$-stable C*-algebras are point-norm dense in the set of all unital embeddings, and that every unital embedding between $\mathcal {D}$-stable C*-algebras is approximately unitarily equivalent to a $\mathcal {D}$-stable embedding. Examples are provided.

We consider the range inclusion and the diagonalization in the Jordan algebra $\mathcal {S}_C$ of C-symmetric operators, that are, bounded linear operators T satisfying $CTC =T^{*}$, where C is a conjugation on a separable complex Hilbert space $\mathcal H$. For $T\in \mathcal {S}_C$, we aim to describe the set $C_{\mathcal {R}(T)}$ of those operators $A\in \mathcal {S}_C$ satisfying the range inclusion $\mathcal {R}(A)\subset \mathcal {R}(T)$. It is proved that (i) $C_{\mathcal {R}(T)}=T\mathcal {S}_C T$ if and only if $\mathcal {R}(T)$ is closed, (ii) $\overline {C_{\mathcal {R}(T)}}=\overline {T\mathcal {S}_C T}$, and (iii) $C_{\overline {\mathcal {R}(T)}}$ is the closure of $C_{\mathcal {R}(T)}$ in the strong operator topology. Also, we extend the classical Weyl–von Neumann Theorem to $\mathcal {S}_C$, showing that every self-adjoint operator in $\mathcal {S}_C$ is the sum of a diagonal operator in $\mathcal {S}_C$ and a compact operator with arbitrarily small Schatten p-norm for $p\in (1,\infty )$.

We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II$_1$ factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C$^*$-dynamics. Given a countable discrete group G and an amenable action $G\curvearrowright M$ on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.

We discuss a strategy for classifying anomalous actions through model action absorption. We use this to upgrade existing classification results for Rokhlin actions of finite groups on C$^*$-algebras, with further assuming a UHF-absorption condition, to a classification of anomalous actions on these C$^*$-algebras.

We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.

Let A be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.

In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$-theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

We introduce the notion of weakly local reflexivity in operator space theory and prove that any dual operator space is weakly locally reflexive.