Pure States on Cuntz Algebras Arising from Geometric Progressions

Let \(\mathcal {O}_{n}\) denote the Cuntz algebra for n ≥ 2. We introduce an embedding f of \(\mathcal {O}_{m}\) into \(\mathcal {O}_{n}\) arising from a geometric progression of Cuntz generaters of \(\mathcal {O}_{n}\). By identifying \(\mathcal {O}_{m}\) with \(f(\mathcal {O}_{m})\), we extend Cuntz states on \(\mathcal {O}_{m}\) to \(\mathcal {O}_{n}\). We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of all such extensions up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing state into a convex hull of pure states. The complete set of invariants of all GNS representations by such pure states is given as a certain set of complex unit vectors.     

On square-increasing ordered monoids and idempotent semirings

Let \(\mathcal {V}\) be the variety of square-increasing idempotent semirings. Its members can be viewed as semilattice-ordered monoids satisfying \(x\le x^{2}\). We show that the universal theory of \(\mathcal {V}\) is decidable. In order to prove this result, we investigate the class \(\mathcal {Q}\) whose members are ordered-monoid subreducts of members from \(\mathcal {V}\). In particular, we prove that finitely generated members from \(\mathcal {Q}\) are well-partially-ordered and residually finite.     

Quadratic programs with hollows

Let \(\mathcal {F}\) be a quadratically constrained, possibly nonconvex, bounded set, and let \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) denote ellipsoids contained in \(\mathcal {F}\) with non-intersecting interiors. We prove that minimizing an arbitrary quadratic \(q(\cdot )\) over \(\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)\) is no more difficult than minimizing \(q(\cdot )\) over \(\mathcal {F}\) in the following sense: if a given semidefinite-programming (SDP) relaxation for \(\min \{ q(x) : x \in \mathcal {F}\}\) is tight, then the addition of l linear constraints derived from \(\mathcal {E}_1, \ldots , \mathcal {E}_l\) yields a tight SDP relaxation for \(\min \{ q(x) : x \in \mathcal {G}\}\). We also prove that the convex hull of \(\{ (x,xx^T) : x \in \mathcal {G}\}\) equals the intersection of the convex hull of \(\{ (x,xx^T) : x \in \mathcal {F}\}\) with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.     

Shadows of the axiom of choice in the universe $$L(\mathbb {R})$$

We show that several theorems about Polish spaces, which depend on the axiom of choice (\(\mathcal {AC}\)), have interesting corollaries that are theorems of the theory \(\mathcal {ZF} + \mathcal {DC}\), where \(\mathcal {DC}\) is the axiom of dependent choices. Surprisingly it is natural to use the full \(\mathcal {AC}\) to prove the existence of these proofs; in fact we do not even know the proofs in \(\mathcal {ZF} + \mathcal {DC}\). Let \(\mathcal {AD}\) denote the axiom of determinacy. We show also, in the theory \(\mathcal {ZF} + \mathcal {AD} + V = L(\mathbb {R})\), a theorem which strenghtens and generalizes the theorem of Drinfeld (Funct Anal Appl 18:245–246, 1985) and Margulis (Monatshefte Math 90:233–235, 1980) about the unicity of Lebesgue’s measure. This generalization is false in \(\mathcal {ZFC}\), but assuming the existence of large enough cardinals it is true in the model \(\langle L(\mathbb {R}),\in \rangle \).     

Branching Rules for Finite-Dimensional $\mathcal {U}_{q}(\mathfrak {su}(3))$-Representations with Respect to a Right Coideal Subalgebra

We consider the quantum symmetric pair \((\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})\) where \(\mathcal {B}\) is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of \(\mathcal {B}\) are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of \(\mathcal {U}_{q}(\mathfrak {su}(3))\) to \(\mathcal {B}\) decomposes multiplicity free into irreducible representations of \(\mathcal {B}\). Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.     

Approximation of Invariant Subspaces in Some Dirichlet-Type Spaces

The Hilbert space \(\mathcal {D}_{2}\) is the space of all holomorphic functions f defined on the open unit disc \(\mathbb {D}\) such that \({f}^{'}\) is in the Hardy Hilbert space \(\mathbf {H}^2.\) In this paper, we prove that the invariant subspaces of \(\mathcal {D}_{2}\) with respect to multiplication operator \(M_{z}\) can be approximated with finite co-dimensional invariant subspaces. We also obtain a partial result in this direction for the classical Dirichlet space.     

Relative Transpose and its Dual with Respect to a Bimodule

For a class of modules \(\mathcal {X}\), we introduce the \(\mathcal {X}\)-transpose of a module with respect to a bimodule, which unifies some well-known transposes. Let \(\mathcal {V}\) be a subclass of \(\mathcal {X}\). The relations between \(\mathcal {X}\)-transposes and \(\mathcal {V}\)-transposes are investigated under the condition that \(\mathcal {V}\) is a generator or cogenerator of \(\mathcal {X}\). The dual aspects of \(\mathcal {X}\)-transposes are also discussed. Then we give some applications of these results. In particular, the dual counterparts of Gorenstein transposes are established.     

Maps with the Unique Extension Property and $$\hbox {C}^{*}$$-Extreme Points

We investigate boundary representations in the context where Hilbert spaces are replaced by \(\hbox {C}^{*}\)-modules over abelian von Neumann algebras and apply this to study \(\hbox {C}^{*}\)-extreme points. We present an (unexpected) example of a weak* compact \(\mathcal {B}\)-convex subset of \({\mathbb {B}}(\mathcal {H})\) without \(\mathcal {B}\)-extreme points, where \(\mathcal {B}\) is an abelian von Neumann algebra on a Hilbert space \(\mathcal {H}\). On the other hand, if \(\mathcal {A}\) is a von Neumann algebra with a separable predual and whose finite part is injective, we show that each weak* compact \(\mathcal {A}\)-convex subset of \(\ell ^{\infty }(\mathcal {A})\) is generated by its \(\mathcal {A}\)-extreme points.     

Local maximality for bounded plurifinely plurisubharmonic functions

In this paper, we prove that \(\mathcal {F}\)-maximality is an \(\mathcal {F}\)-local notion for bounded \(\mathcal {F}\)-plurisubharmonic functions.     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

The deformation of symplectic critical surfaces in a Kähler surface-II—compactness

In this paper we consider the compactness of \(\beta \)-symplectic critical surfaces in a Kähler surface. Let M be a compact Kähler surface and \(\Sigma _i\subset M\) be a sequence of closed \(\beta _i\)-symplectic critical surfaces with \(\beta _i\rightarrow \beta _0\in (0,\infty )\). Suppose the quantity \(\int _{\Sigma _i}\frac{1}{\cos ^q\alpha _i}d\mu _i\) (for some \(q>4\)) and the genus of \(\Sigma _{i}\) are bounded, then there exists a finite set of points \({{\mathcal {S}}}\subset M\) and a subsequence \(\Sigma _{i'}\) which converges uniformly in the \(C^l\) topology (for any \(l<\infty \)) on compact subsets of \(M\backslash {{\mathcal {S}}}\) to a \(\beta _0\)-symplectic critical surface \(\Sigma \subset M\), each connected component of \(\Sigma \setminus {{\mathcal {S}}}\) can be extended smoothly across \({{\mathcal {S}}}\).     

Positive Systems of Kostant Roots

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F₄,E₆,E₇, and E₈. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions \(u(x)\ge 0\) to the semilinear diffusion equation \(\mathcal {L}u= f(u)\), posed in a bounded domain \(\Omega \subset \mathbb {R}^N\) with appropriate homogeneous Dirichlet or outer boundary conditions. The operator \(\mathcal {L}\) may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian \((-\Delta )^s\) (\(0<s<1\)) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function \(f(u)\sim u^p\), with \(p\le 1\). The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number \(\frac{2s}{1-p}\) goes below the exponent \(\gamma \in (0,1]\) corresponding to the Hölder regularity of the first eigenfunction \(\mathcal {L}\Phi _1=\lambda _1 \Phi _1\). Indeed a change of boundary regularity happens in the different regimes \(\frac{2s}{1-p} \gtreqqless \gamma \), and in particular a logarithmic correction appears in the “critical” case \(\frac{2s}{1-p} = \gamma \). For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range \(0<s\le (1-p)/2\).     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Ranks of ideals in inverse semigroups of difunctional binary relations

The set \(\mathcal {D}_n\) of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (Publ Math Debrecen 78(2):253–282, 2011), which asks: What is the rank (smallest size of a generating set) of \(\mathcal {D}_n\)? Specifically, we show that the rank of \(\mathcal {D}_n\) is \(B(n)+n\), where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of \(\mathcal {D}_n\). Although \(\mathcal {D}_n\) bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of \(\mathscr {J}\)-classes), we note that the fast growth of \({\text {rank}}(\mathcal {D}_n)\) as a function of n is a property not shared with these other families.     

A partition relation for pairs on $$\omega ^{\omega ^\omega }$$

We consider colorings of the pairs of a family \(\mathcal {F}\subseteq {{\mathrm{FIN}}}\) of topological type \(\omega ^{\omega ^k}\), for \(k>1\); and we find a homogeneous family \(\mathcal {G}\subseteq \mathcal {F}\) for each coloring. As a consequence, we complete our study of the partition relation \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}\) identifying \(\omega ^{\omega ^\omega }\) as the smallest ordinal space \(\alpha <\omega _1\) satisfying \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

Some new expansions for certain truncated <Emphasis Type="Italic">q</Emphasis>-series

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.     

On counting subring-submodules of free modules over finite commutative frobenius rings

Let \(\texttt {R}\) be a finite commutative Frobenius ring and \(\texttt {S}\) a Galois extension of \(\texttt {R}\) of degree m. For positive integers k and \(k'\), we determine the number of free \(\texttt {S}\)-submodules \(\mathcal {B}\) of \(\texttt {S}^\ell \) with the property \(k=\texttt {rank}_\texttt {S}(\mathcal {B})\) and \(k'=\texttt {rank}_\texttt {R}(\mathcal {B}\cap \texttt {R}^\ell )\). This corrects the wrong result (Bill in Linear Algebr Appl 22:223–233, 1978, Theorem 6) which was given in the language of codes over finite fields.