In this paper we formulate and solve Nevanlinna-Pick and Carathéodory type problems for tensor algebras with data given on the $N$ -dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.

In this paper, we study the modularity of certain elliptic surfaces by determining their $L$ -series through their monodromy groups.

Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$ . We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$ , for a suitable invertible or unitary $T$ and $\xi \,\in \,\{1,\,-1\}$ , where ${{X}^{t}}$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$ . Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I\,+\,K$ , where $\alpha$ is a scalar and $K$ is compact.

We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple ${{C}^{*}}$ -algebras. The first result of this kind was shown by Powers for the $\text{UHF}$ algbras some 30 years ago.

Recently there has been tremendous interest in counting the number of integral points in $n$ -dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$ -dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$ -dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of ${{\mathbb{C}}^{n}},\,n\,>\,1$ . We first give density conditions for a sequence to be interpolating for the class ${{A}^{-\infty }}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n\,=\,1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

The $abc$ -conjecture is applied to various questions involving the number of distinct fields $\mathbb{Q}\left( \sqrt{f(n)} \right)$ , as we vary over integers $n$ .

We study the farthest-point distance function, which measures the distance from $z\,\in \,\mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane.

The logarithm of this distance is subharmonic as a function of $z$ , and equals the logarithmic potential of a unique probability measure with unbounded support. This measure ${{\sigma }_{E}}$ has many interesting properties that reflect the topology and geometry of the compact set $E$ . We prove ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$ -gon for some odd $n$ . Also we show ${{\sigma }_{E}}(E)\,=\,\frac{1}{2}$ for smooth convex sets of constant width. We conjecture ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for all $E$ .

This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case.

We show that the multi-sided inclusion ${{R}^{\otimes l}}\,\subset \,R$ of braid-type subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$ , introduced in Multi-sided braid type subfactors $[\text{E}3]$ , contains a sequence of intermediate subfactors: ${{R}^{\otimes l\,}}\subset \,{{R}^{\otimes l-1\,}}\subset \,\,\cdots \,\,\subset \,{{R}^{\otimes 2\,}}\subset \,R$ . That is, every $t$ -sided subfactor is an intermediate subfactor for the inclusion ${{R}^{\otimes l}}\,\subset \,R,\,\text{for 2}\,\le \,t\,\le \,l$ . Moreover, we also show that if $t\,>\,m$ then ${{R}^{\otimes t}}\,\subset \,{{R}^{\otimes m}}$ is conjugate to ${{R}^{\otimes t-m+1\,}}\subset \,R$ . Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.

It is known that a unital simple ${{C}^{*}}$ -algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital ${{C}^{*}}$ -algebras with tracial topological rank zero that have real rank other than zero.

Let $0\,\to \,J\,\to \,E\,\to A\,\to \,0$ be a short exact sequence of ${{C}^{*}}$ -algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a ${{C}^{*}}$ -algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

We study the dimension of “random” Euclidean sections of direct sums of normed spaces. We compare the obtained results with results from $[\text{LMS}]$ , to show that for the direct sums the standard randomness with respect to the Haar measure on Grassmanian coincides with a much “weaker” randomness of “diagonal” subspaces (Corollary 1.4 and explanation after). We also add some relative information on “phase transition”.

This paper shows that some characterizations of minimally thin sets connected with a domain having smooth boundary and a half-space in particular also hold for the minimally thin sets at a corner point of a special domain with corners, i.e., the minimally thin set at $\infty$ of a cone.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$ -space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

We show that if $\mathcal{A}$ is a class of ${{C}^{*}}$ -algebras for which the set of formal relations $\mathcal{R}$ is weakly stable, then $\mathcal{R}$ is weakly stable for the class $B$ that contains $\mathcal{A}$ and all the inductive limits that can be constructed with the ${{C}^{*}}$ -algebras in $\mathcal{A}$ .

A set of formal relations $\mathcal{R}$ is said to be weakly stable for a class $\mathcal{C}$ of ${{C}^{*}}$ -algebras if, in any ${{C}^{*}}$ -algebra $A\,\in \,\mathcal{C}$ , close to an approximate representation of the set $\mathcal{R}$ in $A$ there is an exact representation of $\mathcal{R}$ in $A$ .

The celebrated theorem of Picard asserts that each non-constant entire function assumes every value infinitely often, with at most one exception. The Riemann zeta function has this Picard behaviour in a sequence of discs lying in the critical band and whose diameters tend to zero. According to the Riemann hypothesis, the value zero would be this (unique) exceptional value.

We determine when an automorphism of a subfactor of type $\text{II}{{\text{I}}_{0}}$ with finite index is nonstrongly free in the sense of $\text{C}$ . Winsløw in terms of the modular endomorphisms introduced by M. Izumi.

We consider a class $(A,\,S,\,\alpha )$ of dynamical systems, where $S$ is an Ore semigroup and $\alpha$ is an action such that each ${{\alpha }_{s}}$ is injective and extendible (i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$ . We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group $G$ of $S$ constructed in [15, Proposition 6.1]. It turns out that the full crossed product by this coaction is isomorphic to $A\,{{\rtimes }_{\alpha }}\,S$ . If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

Let $M$ be a compact, connected, orientable, irreducible 3-manifold with a torus boundary. It is known that if two Dehn fillings on $M$ along the boundary produce a reducible manifold and a manifold containing a Klein bottle, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic zero and ${{V}_{*}}$ be a space of linear functionals on $V$ which separate the points of $V$ . We consider $V\,\otimes \,{{V}_{*}}$ as a Lie algebra of finite rank operators on $V$ , and set $\mathfrak{g}\mathfrak{l}(V,\,{{V}_{*}})\,:=\,V\,\otimes \,{{V}_{*}}$ . We define a Cartan subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ under the assumption that $\mathbb{K}$ is algebraically closed. A subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ is a Cartan subalgebra if and only if it equals ${{\oplus }_{j}}({{V}_{j}}\,\otimes {{({{V}_{j}})}_{*}})\,\oplus \,({{V}^{0}}\,\otimes \,V_{*}^{0})$ for some one-dimensional subspaces ${{V}_{j}}\subseteq V$ and ${{\text{(}{{V}_{j}}\text{)}}_{*}}\subseteq {{V}_{*}}$ with ${{({{V}_{i}})}_{*}}({{V}_{j}})\,=\,{{\delta }_{ij}}\mathbb{K}$ and such that the spaces $V_{*}^{0}=\bigcap{_{j}}{{({{V}_{j}})}^{\bot }}\subseteq {{V}_{*}}$ and ${{V}^{0}}=\bigcap{_{j}}{{\left( {{({{V}_{j}})}_{*}} \right)}^{\bot }}\subseteq V$ satisfy $V_{*}^{0}({{V}^{0}})\,=\,\{0\}$ . We then discuss explicit constructions of subspaces ${{V}_{j}}$ and ${{({{V}_{j}})}_{*}}$ as above. Our second main result claims that a Cartan subalgebra of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra $\mathfrak{h}$ which coincides with the maximal locally nilpotent $\mathfrak{h}$ -submodule of $\mathfrak{g}\mathfrak{l}(V,{{V}_{*}})$ , and such that the adjoint representation of $\mathfrak{h}$ is locally finite.