It is shown that any finite-dimensional homomorphic image of an inverse limit of nilpotent not-necessarily-associative algebras over a field is nilpotent. More generally, this is true of algebras over a general commutative ring $k$, with “finite- dimensional” replaced by “of finite length as a $k$-module.”

These results are obtained by considering the multiplication algebra $M(A)$ of an algebra $A$ (the associative algebra of $k$-linear maps $A → A$ generated by left and right multiplications by elements of $A$), and its behavior with respect to nilpotence, inverse limits, and homomorphic images.

As a corollary, it is shown that a finite-dimensional homomorphic image of an inverse limit of finite-dimensional solvable Lie algebras over a field of characteristic 0 is solvable.

It is also shown by example that infinite-dimensional homomorphic images of pro-nilpotent algebras can have properties far from those of nilpotent algebras; in particular, properties that imply that they are not residually nilpotent.

Several open questions and directions for further investigation are noted.

Let $\mathbf{V}$ be a variety of not necessarily associative algebras, and $A$ an inverse limit of nilpotent algebras $A_i\in\mathbf{V}$, such that some finitely generated subalgebra $S\subseteq A$ is dense in $A$ under the inverse limit of the discrete topologies on the $A_i$.

A sufficient condition on $\mathbf{V}$ is obtained for all algebra homomorphisms from $A$ to finite-dimensional algebras $B$ to be continuous; in other words, for the kernels of all such homomorphisms to be open ideals. This condition is satisfied, in particular, if $\mathbf{V}$ is the variety of associative, Lie, or Jordan algebras.

Examples are given showing the need for our hypotheses, and some open questions are noted.

We describe the possible Mordell–Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in P(2, 3, 1, 1, 1) that admit no variation of Hodge structure (Remark 9.3).

Let $Y := \operatorname{Hom}(\mathbb{Z}^n, \operatorname{SU}(2))$ denote the space of commuting $n$-tuples in $\operatorname{SU}(2)$. We determine the homotopy type of the suspension $\Sigma Y$, and compute the integral cohomology groups of $Y$ for all positive integers $n$.

An effective computation of the kernels of the chaos decomposition of Lévy functionals is used, to prove, among other things, a chain- and product-rule of the Malliavin derivative for a large class of Lévy processes. In case of finite and infinite-dimensional Brownian motion, the well-known rules are obtained, but for Poisson processes, the results are new. The kernels of a Lévy functional can be computed by taking the expected value of the product of this functional and multiple white noise of the Lévy process.

In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in $n$-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in certain $n$-manifolds.

The structure of the $C^*$-algebras corresponding to even-dimensional mirror quantum spheres is investigated. It is shown that they are isomorphic to both Cuntz-Pimsner algebras of certain $C^*$-correspondences and $C^*$-algebras of certain labelled graphs. In order to achieve this, categories of labelled graphs and $C^*$-correspondences are studied. A functor from labelled graphs to $C^*$-correspondences is constructed, such that the corresponding associated $C^*$-algebras are isomorphic. Furthermore, it is shown that $C^*$-correspondences for the mirror quantum spheres arise via a general construction of restricted direct sum.

We consider the Szegö kernel for domains in ${\mathbb C}^{2}$ given by \[ \Omega = \bigl\{(z,w) : \mathrm{Im} (w) > b\bigl(\mathrm{Re} (z)\bigr)\bigr\} \] for $b$ a non-convex quartic polynomial with positive leading coefficient. Such domains are not pseudoconvex. We describe a subset of $\overline{\Omega} \times \overline{\Omega}$ on which the kernel and all of its derivatives are finite. We show that there are points off the diagonal of $\partial \Omega \times \partial \Omega$ at which the Szegö kernel is infinite as well as points on the diagonal at which it is finite.

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gröbner basis. We give a lower bound on $d$ such that the defining ideal of $d$th Veronese subring admits a quadratic Gröbner basis. Eisenbud–Reeves–Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud–Reeves–Totaro’s lower bound.

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a $p$-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of $p$-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, embeddings of chains and antichains into the Tukey types, and Tukey types generated by block-basic ultrafilters on FIN.

We compare algebraic objects related to a compact quantum group action on a unital $C^∗$-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.

We construct a non-atomic strong operator topology-dense probability measure on the set of unitary operators acting on a separable Hilbert space, such that the $C^∗$-algebra generated by $n ≥ 3$ independently chosen random unitaries is almost surely non-exact.

This paper deals with three issues: (1) Unitary representations $U$ of a scale of (finite and infinite dimensional) non-compact Lie groups $G(H)$ built on a fixed complex Hilbert space $H$; and their covariant systems. Our computations for these representations make use of the associated Lie algebras. (2) The covariant representations involve the $C^{*}$-algebras going by the names, the Toeplitz algebras, and the Cuntz algebras. (3) An essential result which also is used throughout is our computation of the commutant of the unitary representation $U$ of $G(H)$ mentioned in (1). For a fixed Hilbert space $H$, we apportion the commutant as a specific projective limit-algebra of operators.

We study the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces. The separation of the construction pieces is closely related to the Hausdorff measure of the corresponding limit set. In particular, we investigate different separation conditions for semiconformal iterated function systems. Our work generalizes well-known results on self-similar sets in metric spaces as well as results on controlled Moran constructions in Euclidean spaces.

A generalization of the Pistone–Sempi argument, demonstrating the utility of noncommutative Orlicz spaces, is presented. In particular, regular quantum statistical systems are described. The question of lifting positive maps defined on von Neumann algebra to maps on corresponding noncommutative Orlicz spaces is discussed. In particular, we describe those Jordan ∗-morphisms on semifinite von Neumann algebras which in a canonical way induce quantum composition operators on noncommutative Orlicz spaces. Consequently, it is proved that the framework of noncommutative Orlicz spaces is well suited for an analysis of a large class of interesting noncommutative dynamical systems.

The author establishes some geometric criteria for a domain of ${\mathbb R}^n$ with $n\ge2$ to support a $(pn/(n-ps),p)_s$-Hajłasz–Sobolev–Poincaré imbedding with $s\in(0,1]$ and $p\in(n/(n+s), n/s)$ or an $s$-Hajłasz–Trudinger imbedding with $s\in(0,1]$.

We show that a nonvanishing analytic function on a sub-disc of the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the sub-disc. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.

We prove that the condenser equilibrium energy is a superharmonic function when one of the plates of a condenser moves under a holomorphic motion and we characterize the cases where harmonicity occurs by showing that this happens if and only if the equilibrium measure on the moved plate is invariant under the holomorphic motion and the equilibrium measure on the fixed plate is unaffected. Also, in many cases, we show that harmonicity of the above function occurs only when the holomorphic motion is related with the level sets of the equilibrium potential of the condenser. In the case where the holomorphic motion is a dilation, we prove that harmonicity occurs if and only if the condenser is essentially an annulus with center at the origin.

We consider the problem of classifying (possibly noncommutative) $R$-algebras of low rank over an arbitrary base ring $R$. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard involution. We then investigate a class of exceptional rings of degree 2 which occur in every rank $n ≥ 1$ and show that they essentially characterize all algebras of degree 2 and rank 3.

Using Morse-theoretic techniques, we show that the moduli space of $U^{∗}(2n)$-Higgs bundles over a compact Riemann surface is connected.

Recently, I. Pritsker considered a Bergman-space version of Mahler’s measure, and obtained many nice properties such as the arithmetic nature, relation with asymptotic zero distribution, etc. (Illinois J. Math. 52 (2009) 347–363). In this paper, we define a Fock-space analogue of Mahler’s measure, and show a similar version of Lehmer’s conjecture. Inspired by this result, we establish an equivalent form of Lehmer’s conjecture. Also, this consideration is done on weighted Bergman spaces. However, it is shown that in this case the corresponding form of Lehmer’s conjecture fails. In addition, we give an affirmative answer to an approximation question raised by I. Pritsker (Illinois J. Math. 52 (2009) 347–363).

Speissegger proved that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Here we give a first order version of this result: having introduced the notion of definably complete Baire structure, we define the relative Pfaffian closure of an o-minimal structure inside a definably complete Baire structure, and we prove its o-minimality. We derive effective bounds on some topological invariants of sets definable in the Pfaffian closure of an o-minimal expansion of the real field.

We prove that if a metric space satisfies a suitable growth condition in the quasihyperbolic metric and the Gehring–Hayman theorem in the original metric, then the inner boundary of the space is homeomorphic to the Gromov boundary. Thus, the inner boundary is compact.

Let $N$ be a connected, simply connected, nilpotent Lie group and let $K$ be a compact subgroup of the automorphism group of $N$. We study the density of Schwartz functions in the kernels of $K$-orbits and characterize $K$-prime ideals. For this purpose a retract theory for $K$-actions has to be established.