We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra $\mathcal {M}$ is never model complete if its direct integral decomposition contains $\mathrm {II}_1$ factors $\mathcal {N}$ such that $M_2(\mathcal {N})$ embeds into an ultrapower of $\mathcal {N}$. The proof in the case of $\mathrm {II}_1$ factors uses an explicit construction based on random matrices and quantum expanders.

We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ of a II1 factor, which was so far only known for totally disconnected groups, compact groups, and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as $\operatorname{Out} M$.

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.

For a state $\omega$ on a C$^{*}$-algebra $A$, we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \varphi$, where $\varphi$ is a map on $A$ of the form $\varphi (x)=\sum \nolimits _{i=1}^{n}a_i^{*}xa_i,$ $\sum \nolimits _{i=1}^{n}a_i^{*}a_i=1$ ($a_i\in A$, $n\in \mathbb {N}$). These are precisely the states $\rho$ that satisfy $\|\rho |J\|\leq \|\omega |J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $\mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$, where $\psi$ is a quantum channel on $\mathcal {R}$ (that is, a map of the form $\psi (x)=\sum \nolimits _ja_j^{*}xa_j$, where $a_j\in \mathcal {R}$ are such that the sum $\sum \nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^{*}$-algebra and for properly infinite von Neumann algebras the converse also holds.

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.

We study class 𝒮 for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class 𝒮 and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class 𝒮 is a measure equivalence invariant.

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusions $L(H)\subset L(G)$.

In this paper, we give a complete description of left symmetric points forBirkhoff orthogonality in the preduals of von Neumann algebras. As aconsequence, except for $\mathbb{C}$ , $\ell _{\infty }^{2}$ and $M_{2}(\mathbb{C})$ , there are no von Neumann algebras whose preduals havenonzero left symmetric points for Birkhoff orthogonality.

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

We give a short proof of a result of T. Bates and T. Giordano stating that any uniformly bounded Borel cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. We also point out a separability issue in their proof. Our approach is based on the existence of a non-positive curvature metric on the positive cone of a finite von Neumann algebra.

We show that if $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ has the weak Haagerup property, then both $M$ and $\unicode[STIX]{x1D6E4}$ have the weak Haagerup property, and if $\unicode[STIX]{x1D6E4}$ is an amenable group, then the weak Haagerup property of $M$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ . We also give a condition under which the weak Haagerup property for $M$ and $\unicode[STIX]{x1D6E4}$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ .

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan maximal abelian self-adjoint subalgebras (MASAs) using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman and Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group $\mathbb{G}$ a locally compact group $\tilde{\mathbb{G}}$ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$ . This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde{\mathbb{G}}$ the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$ .

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal{M}$ can be computed from the decomposability number, $\text{dens}\left( \mathcal{M} \right)$ , and the minimal cardinality of a generating set, $\text{gen}\left( \mathcal{M} \right)$ . Applications include the equivalence of the well-known generator problem, “Is every separably-acting von Neumann algebra singly-generated?”, with the formally stronger questions, “Is every countably-generated von Neumann algebra singly-generated?” and “Is the gen invariant monotone?” Modulo the generator problem, we determine the range of the invariant $\left( \text{gen}\left( \mathcal{M} \right),\,\text{dens}\left( \mathcal{M} \right) \right)$ , which is mostly governed by the inequality $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$ .

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).

Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂn×n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂn×n) is shown to be undecidable.

We introduce a new topology, weaker than the gap topology, on the space of self-adjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of self-adjoint Breuer–Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips's analytical definition and derive integral formulae for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey and Phillips. Furthermore, we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.

In this paper we prove algebraic generalizations of some results of C. J. K. Batty and A. B. Thaheem, concerned with the identity α + α−1 = β + β−1 where α and β are automorphisms of a C*-algebra. The main result asserts that if automorphisms α, β of a semiprime ring R satisfy α + α-1 = β + β−1 then there exist invariant ideals U1, U2 and U3 of R such that Ui ∩ Uj = 0, i ≠ j, U1 ⊕ U2 ⊕ U3 is an essential ideal, α = β on U1, α = β−1 on U2, and α2 = β2 = α−2 on U3. Furthermore, if the annihilator of any ideal in R is a direct summand (in particular, if R is a von Neumann algebra), then U1 ⊕ U2 ⊕ U3 = R.