Papers

We prove an \(L\log L\)-endpoint Cwikel estimate on quantum tori $\mathbb T_\theta^d$, i.e., for every \(d\geq 1\) and \(x\in L\log L(\mathbb T_\theta^d)\), \begin{equation*} (1-\Delta)^{-\frac{d}{4}}\lambda(x)(1-\Delta)^{-\frac{d}{4}}\in\mathcal{L}_{1,\infty} \end{equation*} with a bound depending only on the dimension, where $\lambda(x)$ is the left multiplication by $x$. The main ingredient is an abstract weak Hilbert–Schmidt estimate for multiplication operators in finite von Neumann algebras whose \(L_2\)-spaces admit orthonormal bases of unitaries. When $\theta=0$, our argument also gives a simpler proof of the $L\log L$ Cwikel estimate on classical tori. For $d\geq2$, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for finite sums \[ \sum_{\ell=1}^N S_\ell(1-\Delta)^{-d/4}\lambda(x_\ell) (1-\Delta)^{-d/4}T_\ell, \] where $x_\ell\in L\log L(\mathbb T_\theta^d)$ and $S_\ell,T_\ell\in\Pi$, the $C^*$-algebra generated by $\lambda(C(\mathbb T_\theta^d))$ and Fourier multipliers with continuous symbols homogeneous of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\log L$ potentials.

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $\Omega=\mathbb{D}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $\Omega(1)$ admits a transitive subgroup of automorphisms with NC lifts and $\Omega(1)$ has the bounded $H^\infty$-Gleason property at the origin, then $\|f(T)\|\leq C_{\Omega,n}\|f\|_{H^\infty(\Omega(1))}$ for every commuting tuple $T\in\Omega(n)$, with $C_{\Omega,n}=(2\rho_{\Omega}K_{\Omega})^{n-1}$; here $K_{\Omega}$ is the Gleason constant and $\rho_{\Omega}$ is the sum of the suprema of the coordinate functions on $\Omega(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where $\rho_{\Omega}=d$, yielding an unconditional estimate $(2dK_{\Omega})^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^\infty$-Gleason property at the origin, which we do not establish when $\min(p,q)\geq 2$.

We study maximal subgroups of topological full groups of irreducible one-sided shifts of finite type whose associated groupoids have Cantor unit space. For each wide open subgroupoid of such a groupoid, we construct a subgroupoid graph and identify the subgroupoid with a full compact open reduction of the corresponding countable graph groupoid. Every minimal wide open subgroupoid is either AF or purely infinite. A proper wide open subgroupoid is prime if it admits no proper intermediate wide open subgroupoid. We prove a trichotomy for prime subgroupoids: they are minimal, arise from a clopen bipartition, or have a unique nonempty proper closed invariant set, which is nowhere dense. For a prime subgroupoid, the normalizer of its topological full group is maximal in the ambient topological full group precisely when it maps onto the ambient abelianization. An analogous criterion holds in every subgroup containing the ambient commutator subgroup. For a minimal prime subgroupoid, its topological full group is maximal precisely when it equals its normalizer and the inclusion induces a surjection on first groupoid homology. We also describe all fully amplified maximal subgroups, namely those closed under local agreement. For Thompson's group $V$, the stabilizer of every simple type system is maximal, extending results of Belk, Bleak, Quick and Skipper for finite type systems and certain classes of infinite ones. Finally, every subgroup containing the ambient commutator subgroup has continuum many pairwise nonisomorphic maximal subgroups acting minimally on the unit space.

We study operator systems arising from Peter-Weyl and Dirac spectral truncations of $\mathrm{SU}(2)$ in the sense of Connes-Van Suijlekom. We determine operator system invariants for these systems, including the $C^*$-envelope and uniform bounds on the propagation number. For Peter-Weyl truncations, we find $\mathrm{prop} \leq 2$ for all connected compact Lie groups, generalizing known results for the torus. We also carry out an analysis of the spectral localizer by Loring and Schulz-Baldes on $\mathrm{SU}(2)$. In this analysis, we find numerical evidence that an index map for operator systems previously proposed by Van Suijlekom only captures the right information in a suitable interval of the parameter $\kappa$, instead of the limit $\kappa \downarrow 0$.

Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{*}$-algebras generated by free semicircular systems: the isomorphism class of a $C^{*}$-algebra generated by a semicircular $n$-tuple does not depend on $n\in\{2,3,\dots,+\infty\}$. Similarly, reduced free products of $n$ copies of the Jiang–Su algebra are isomorphic for different $n$ in this range. While reduced free group $C^{*}$-algebras of different ranks are not isomorphic (by a result of Pimsner–Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of $C^{*}$- and $W^{*}$-algebras; a relative version of that property was previously used to derive free complementation results.

Let $\Gamma$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,\tau_Q)$ in a trace-preserving way. We study invariant von Neumann subalgebras of $Q\rtimes\Gamma$ and of $L(\Gamma)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto $Q$ is scalar is necessarily contained in $L(\Gamma)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\mathbb{F}_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors.

The modular data of the doubles of the Haagerup–Izumi categories is a grafting of two finite metric groups (conjectured by Evans–Gannon, proved for untwisted categories by Gannon–Schopieray–Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ $S$-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy $\varphi$, a word in $L$ and $R$, the two groups are $A_\mp=\coker(\varphi\mp1)$, with the torsion linking forms of the mapping tori of $\varphi$ and $-\varphi$. Over all words of trace $m^2+2$ these forms reproduce the Evans–Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon–Schopieray–Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, $c_{\eff}=\#L$, and the handle-gluing values $2^{\#L}\lambda^{\pm1}|A_-||A_+|$ at the identity classes, $\lambda$ the stretch factor. Each kernel vector of $\varphi\mp1$ modulo $|A_\mp|$ at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of $\sigma V^*$, $c\equiv\#L-6\#R\pmod{24}$.

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

We introduce a class of extensions of unital operator systems satisfying a certain Toeplitz type property. We show that these extensions induce corresponding extensions of the $C^*$-algebras they generate. We prove that hyperrigidity is preserved under these extensions, yielding new examples through iteration. We introduce the quantum double suspension of operator systems as a special class of these extensions and show that its iterates yield the Hong-Szymański quantum double suspensions at the $C^*$-algebra level, encompassing, in particular, the Vaksman-Soibelman quantum spheres. We construct hyperrigid operator systems underlying these quantum spheres. As a further application, we show that every counterexample to Arveson's hyperrigidity conjecture gives rise to a sequence of counterexamples. We identify the $C^*$-envelopes of these extensions and use this identification to obtain a $K$-theoretic condition ensuring that each such sequence contains infinitely many pairwise non-isomorphic counterexamples. We also observe that the validity of the conjecture in the commutative setting is linked to its validity for an infinite family of noncommutative operator systems obtained through successive extensions. Finally, we compute the propagation number, an invariant of operator systems introduced by Connes and van Suijlekom, for these extensions.

Let $2<p<\infty$. We prove that every infinite-dimensional closed subspace $F$ of a noncommutative $L_p$-space contains a subspace completely isomorphic to $\ell_p$, $C_p$, $R_p$, or $C_p\cap R_p$ and completely complemented in $F$. This answers the question of Junge and Xu on the existence of completely symmetric basic sequences.

For every von Neumann algebra \(\M\), we prove that each bounded sequence in the Haagerup space \(L_1(\M)=\M_*\) admits either a norm-convergent subsequence or a \(2\)-co-lacunary subsequence. This answers an open problem of Parcet and Randrianantoanina \cite[Remark~3.11]{ParcetRandrianantoanina2006} and removes both hyperfiniteness and semifiniteness.

We completely classify aperiodic automorphisms of the Razak–Jacelon algebra $\mathcal{W}$ up to outer conjugacy. As a technical ingredient, based on the works of Connes, Herman–Ocneanu, Kishimoto, and Sato, we introduce a notion of weak stability for automorphisms of simple separable monotracial C$^*$-algebras. We also consider automorphisms of UHF algebras of infinite type and the Jiang–Su algebra. In particular, we classify aperiodic “UHF-absorbing” automorphisms of UHF algebras of infinite type up to outer conjugacy.

We examine various applications of logic to $\mathrm{AW}^*$-algebras, particularly the monotone completeness conjecture. We develop a model theory for $\mathrm{AW}^*$-algebras equipped with a faithful normalized quasitrace. Using this language, we provide a separable coding which allows us to reduce the finite case of the monotone completeness conjecture to monotone sequential completeness of $\mathrm{II}_1$ $\mathrm{AW}^*$-factors. Combining an explicit recursive axiomatization with the completeness theorem for continuous logic, we show that this case of the conjecture, and Kaplansky's problem whether every $\mathrm{AW}^*$-factor of type $\mathrm{II}_1$ is a von Neumann algebra, are each equivalent in ZFC to a $\Pi^0_2$ sentence of arithmetic, so that their answers are the same in every $\omega$-model of ZFC. We also consider real $\mathrm{AW}^*$-algebras. Here the Baer-$*$-ring notion of an abelian projection must be distinguished from commutativity; following the real-operator-algebra terminology we call the former semi-abelian. Handelman's matrix theorem implies that a real $\mathrm{AW}^*$-algebra with no nonzero central semi-abelian projection has $\mathrm{AW}^*$ complexification. We answer a number of other open problems in real $\mathrm{AW}^*$-algebras.

Using the recent single-generation theorem for von Neumann $\mathrm{II}_1$ factors with separable predual, we prove the following results on the generators of a separable unital C*-algebra. Let $A$ be a separable unital C*-algebra, and suppose that $\partial \mathrm{T}(A)$ is nonempty, compact, and zero-dimensional. Then (1) If $A$ has strict comparison of positive elements by tracial states, then $A$ can be generated by two ordinary elements. (2) If $A$ has real rank zero and $A$ has strict comparison of projections by tracial states, then $A$ is singly generated. In this case, its single generators form a norm-dense $G_\delta$ subset of $A$. (3) If $A$ has strict comparison of positive elements by tracial states and there is a unitary $u\in A$ such that for every $\tau\in \partial \mathrm{T}(A)$, (a): the spectral distribution of $u$ with respect to $\tau$ is atomless and has full support on $\mathbb T$, and (b): for every $a\in A$, $$\lim_{t \to 0}\sup_{p\in D_\tau, 0<\tau(p)\leq t} \| p\bigl(\pi_\tau(a)-\mathbb E_{D_\tau}(\pi_\tau(a))\bigr)p\|=0, $$ where $M_\tau=\pi_\tau(A)''$, $D_\tau=\pi_\tau(u)''$, $\mathbb E_{D_\tau}: M_\tau \to D_\tau$ is the trace-preserving conditional expectation, and $\pi_\tau$ is the GNS representation associated to $\tau$, then $A$ is singly generated. In particular, Conclusion (3) above implies that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb F_n)$ is singly generated for every $n\in\{2,3,\ldots,\infty\}$.

We present an $\infty$-categorical refinement of classical $KK$-theory and $E$-theory for graded, $G$-equivariant for a locally compact second countable group $G$, Real $C^*$-algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$-theory spectra, and explicitly analyse the induced endomorphisms on $K_0$-groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$- and $E$-categories are not a Dwyer-Kan localization of graded $C^*$-algebras.

We study endpoint Schatten class properties of commutators $[I_\theta^\epsilon,M_x]$ on quantum tori of dimension $d\geq2$, where $I_\theta^\epsilon$ is the Riesz potential of order $\epsilon$ and $M_x$ denotes the left multiplication operator by an operator $x$ on quantum tori $\mathbb T_\theta^d$. For $-\frac{d}{2}<\epsilon<1$, $\epsilon\neq0$, and $x\in L_1(\mathbb T_\theta^d)$, we prove that $[I_\theta^\epsilon,M_x]\in \mathcal L_{\frac{d}{1-\epsilon},\infty}$ if and only if $x\in\dot W_{\frac{d}{1-\epsilon}}^1(\mathbb T_\theta^d)$. We also determine the singular value asymptotics of this commutator for $x\in\dot W_{\frac{d}{1-\epsilon}}^1(\mathbb T_\theta^d)$. The necessary Sobolev condition holds throughout $1-d<\epsilon<1$, $\epsilon\neq0$. For $\epsilon\leq-\frac{d}{2}$, we characterise Hilbert–Schmidt membership, with a logarithmically weighted Fourier condition at $\epsilon=-\frac{d}{2}$. In order to get the singular value asymptotics of commutators, we prove singular value asymptotics for pseudodifferential operators of arbitrary negative order on quantum tori using $C^*$-algebraic principal symbols. This asymptotic result confirms Conjecture~7.1 of McDonald and Ponge (Adv. Math. 412 (2023), 108815).

Let $G$ be a locally compact, second countable group admitting a Reiter sequence, and suppose that $G$ acts on a von Neumann algebra $\mathcal{M}$ equipped with a semifinite trace $\tau$ via trace-preserving automorphisms. In this paper, we prove a maximal inequality for ergodic averages along such Reiter sequences. Our approach relies on approximating the Reiter sequence by a linear combination of characteristic functions of Følner sequences using the layer-cake representation. Furthermore, we establish the bilateral almost uniform convergence of these ergodic averages. As an application, we deduce a non-commutative Wiener-Wintner type theorem along Reiter sequences.

One main aim of our paper is to formulate noncommutative versions of the Lebesgue dominated convergence theorem in the setting of quantum integration theory on von Neumann algebras. Our starting point is a reformulation of classical dominated convergence in terms of weak* convergence. Using our recent sequential-continuity theorem, we obtain dominated convergence formulations for sequentially normal states and, under domination by a fixed bounded integrable majorant, for sequentially normal weights. Integrable, possibly unbounded, majorants are treated first in the tracial setting, then for arbitrary von Neumann algebras using Haagerup's L1 space, and finally using normal semifinite weights and their GNS representations. For any faithful normal semifinite weight $\phi$, weak* convergence of norm-bounded sequences is characterized by `integrating against' bounded positive integrable majorants. We also study noncommutative almost uniform and local almost uniform convergence. In particular, for norm-bounded sequences, local $\phi$-bilaterally almost uniform convergence implies weak* convergence and L1 convergence to 0 of $h^{1/2}(x_n-x)h^{1/2}$ for every positive Haagerup L1 density h, without assuming sigma-finiteness. We also prove a dominated convergence theorem in the predual: fixed positive order domination and Petz's almost uniform convergence imply convergence in the predual norm. We establish implications and equivalences among these convergence notions, connecting dominated convergence with tracial and nontracial integration.

Given $1 < p < \infty$ and a function $m \colon \mathbf{N}_0 \to \mathbf{C}$, we provide a characterization of radial Schur multipliers $M(x,y) = m(d(x,y))$ on the Schatten $p$-class of a tree $(X,d)$ equipped with its graph metric. That is, the operator norm of the radial Schur multiplier $S_M$ is controlled by the norm of the Fourier multiplier $T_m$ on the torus: \[ \big\| S_M \colon S_p(X) \to S_p(X) \big\|_{\mathrm{cb}} \lesssim \frac{p^4}{(p-1)^2} \, \big\| T_m \colon L_p(\mathbf{T}) \to L_p(\mathbf{T}) \big\|_{\mathrm{cb}}, \] with both quantities being comparable when the tree $X$ has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints $p=1,\infty$ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely $S_p$-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference—a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly $S_p$-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups $\mathrm{PGL}_2(\mathbf{Q}_q)$ and $\mathrm{PSL}_2(\mathbf{R})$ with $\mathrm{K}$-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.

Let $\mathcal B=(B_g)_{g\in G}$ be a saturated Fell bundle over a countable discrete group, equipped with a spectral triple. We give explicit sufficient conditions for lifting this triple to the reduced cross-sectional C\Star-algebra $C_r^*(\mathcal B)$. Using finite projective realizations of the homogeneous Morita equivalence $B_e$-bimodules, we transport the base Dirac operator to the individual fibres and assemble the resulting operators into a fibrewise Dirac operator. We introduce a spectral-regularity condition ensuring bounded commutators and give a sufficient criterion in terms of uniform derivative bounds on smooth factor-system data. Combining the fibrewise operator with a length operator associated with a proper translation-bounded matrix-valued length function yields the lifted spectral triple. When $B_e$ is separable, the lifted triple, with a suitable grading, represents the Kasparov product of the vertical class with the class of the unit-fibre triple. We also establish a summability criterion and sufficient conditions for stability under central cocycle twisting. A range of examples illustrates the scope of the construction, including twisted crossed products, free compact Abelian actions, and Fell bundles with nonfree homogeneous fibres.

We characterize non-commutative polynomials of degree 2 which preserve free infinite divisibility as self-adjoint Lie polynomials of degree 2. This extends previous results which characterized homogeneous quadratic forms only. The proof is based on an involution of colored noncrossing partitions cancelling many terms; positive definiteness of the remaining terms is established by an explicit operator model on Fock space. Examples show that higher order Lie polynomials do not preserve free infinite divisibility.

We study closed-set stabilizers in topological full groups of minimal effective ample groupoids with Cantor unit space. For almost finite or purely infinite groupoids, we extend the maximal stabilizer theorem of Grigorchuk and Vorobets to every subgroup containing the commutator subgroup. For a nonempty closed nowhere dense set, maximality of its stabilizer is equivalent to minimality of its stabilizer action, minimality of the regular reduction, and primeness of a canonical wide open subgroupoid. Without assuming almost finiteness or pure infiniteness, we prove spatial rigidity for stabilizers acting minimally on infinite closed nowhere dense sets. Their isomorphism types determine the ambient Kakutani equivalence class and, when Hausdorff, the effective reduction. We also realize every AF groupoid with compact unit space as the full restriction, with its relative topology, of any minimal effective ample groupoid with Cantor unit space to a closed nowhere dense subset. Combining rigidity and realization yields continuum many pairwise nonisomorphic stabilizers in every subgroup containing the commutator subgroup; these stabilizers are maximal when the ambient groupoid is almost finite or purely infinite. The realization also gives explicit AF subquotients of the ambient reduced groupoid $C^*$-algebra, including the CAR algebra, with compatibility on the canonical diagonal subalgebras.

We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle $S_a = a \otimes 1 + \sum_j A_j \otimes s_j$ with deterministic mean $a$. For every real $z$, $\log \Delta(S_a - z) = -1/2 + \inf_b F_z(b)$, where the infimum is over positive definite matrices $b$ such that the map $X \mapsto bXb - \sum_j A_j X A_j$ is positive semidefinite, and $F_z$ is an explicit geodesically convex function. Both sides equal $-\infty$ exactly when $z$ is an atom of the spectral measure. The formula was proposed, without proof, in a recent post of Speicher. In the centered case $a=0$ it gives a new proof of the Mai-Speicher capacity formula, and by hermitization it yields a finite-dimensional variational formula for the logarithmic potential of the Brown measure of an arbitrary matrix-valued semicircular element. The proof identifies the determinant with the value of the Dyson action at its critical point and rewrites this critical-point problem as a saddle problem in three matrix variables. For a spectral parameter $z + i\varepsilon$ in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor $b_\varepsilon$ in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting $\varepsilon \to 0$ in the optimal values, with no compactness assumption on the minimizers. At points where the boundary value of the Cauchy transform is nondegenerate, the scales $b_\varepsilon$ converge and the infimum is attained at their limit; this minimizer lies on the boundary of the feasible set when $z$ is in the support of the spectral measure and in its interior when $z$ is in the resolvent set.

For a $C^*$-algebra $A$ let $\mathcal I(A)$ be the set of all elements $a+b$ with $a,b\in A_+$, $ab=0$ and $\|a\|=\|b\|=1$. We study the algebraic, metric and topological properties of $\mathcal I(A)$. We describe $\mathcal I(A\oplus B)$ and show that $\mathcal I(A/J)\subseteq\pi(\mathcal I(A))$ for the quotient map $\pi:A\to A/J$. For finite-dimensional $A$ we prove $\mathcal I(A)=\operatorname{Sph}^+_A(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and $\operatorname{Sph}^+_A(E)$ is the set of positive norm-one elements at distance one from every element of $E$; we also show that the analogous statement with the minimal projections of the bidual fails in general. When $\mathcal I(A)\neq\varnothing$ the distance from $1$ to $\mathcal I(A)$ is $0$ or $1$, according to whether $A$ has a nontrivial projection. Moreover, $\mathcal I(A)$ is norm compact if and only if $A$ is finite-dimensional, closedness of $\mathcal I(A)$ passes to hereditary subalgebras, $\mathcal I(A)$ is norm closed for every von Neumann algebra, and the extreme points of $\mathcal I(A)$ are exactly its projections. For a compact, locally path connected space $X$ the set $\mathcal I(C(X))$ is closed, whereas $\mathcal I(\ell_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.

We study two invariants of $C^*$-algebras built from positive zero divisors. First, we associate with a $C^*$-algebra $A$ the graph $\Gamma_+(A)$ whose vertices are the nonzero positive zero divisors of $A$, two vertices being adjacent when they are orthogonal. We show that $\Gamma_+$ of the Calkin algebra is connected of diameter $3$, that $\Gamma_+(B(H))$ is connected of diameter $4$ (for $\dim H\ge3$), that $\Gamma_+(A)$ is connected of diameter at most $4$ for every von Neumann factor $A\neq\mathbb C,M_2(\mathbb C)$, and of diameter at most $3$ if $A$ is not of type~I, whereas $\Gamma_+(M_2(\mathbb C))$ is disconnected. Second, we study the zero-divisor real rank $ZDRR(A)$, obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that $ZDRR(A)\le RR(A)$. For commutative algebras $C(X)$ we prove $ZDRR(C(X))=RR(C(X))=\dim X$ when $X$ is a compact almost-$P$-space, while $ZDRR(C(X))=0$ for separable $X$. Using Levy's connected, totally ordered almost-$P$-space we construct unital commutative $C^*$-algebras $A_n$ with $ZDRR(A_n)=n$ for every $n\in\mathbb N_0$. On the other hand $ZDRR(C(X\times[0,1]))=0$ for every compact Hausdorff space $X$, so $ZDRR$ and $RR$ can differ arbitrarily.

We introduce the notion of essential tracial rank at most one for simple $C^*$-algebras. We show that separable simple $C^*$-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple $C^*$-algebras with this property is $\mathcal Z$-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable $\mathcal Z$-stable $C^*$-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable $C^*$-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable $C^*$-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.

We show that certain classes of unital, simple, separable C*-algebras with comparison are singly generated, including tracially AF algebras and monotracial crossed products of actions on the Cantor space. In particular, we show that simple, separable, tracially AF algebras have generator rank one.

In the context of Fell bundles over discrete groups, we introduce and analyze suitable notions of direct sum and tensor product of actions of Fell bundles on Hilbert bundles. For $C^*$-dynamical systems, where actions arise from equivariant representations, we discuss how the tensor product of two such actions is related to the action associated with the tensor product of the underlying equivariant representations. Likewise, since any action gives rise to a $C^*$-correspondence between the full cross-sectional $C^*$-algebras of the involved Fell bundles, we examine the connection between the $C^*$-correspondence associated with the tensor product of two given actions and the tensor product of the associated $C^*$-correspondences. We also discuss how the tensor product of actions naturally appears in Rieffel-like induction processes. Finally we present related results on the Fourier-Stieltjes and Fourier algebras of a Fell bundle.

We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.

We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.