MATHEMATICA SCANDINAVICA

The Kalton-Peck space as a spreading model

2024-02-12T09:59:37+01:00

The so-called Kalton-Peck space $Z_2$ is a twisted Hilbert space induced, using complex interpolation, by $c_0$ or $\ell _p$ for any $1\leq p\neq 2<\infty $. Kalton and Peck developed a scheme of results for $Z_2$ showing that it is a very rigid space. For example, every normalized basic sequence in $Z_2$ contains a subsequence which is equivalent to either the Hilbert copy $\ell _2$ or the Orlicz space $\ell _M$. Recently, new examples of twisted Hilbert spaces, which are induced by asymptotic $\ell _p$-spaces, have appeared on the stage. Thus, our aim is to extend the Kalton-Peck theory of $Z_2$ to twisted Hilbert spaces $Z(X)$ induced by asymptotic $c_0$ or $\ell _p$-spaces $X$ for $1\leq p<\infty $. One of the novelties is to use spreading models to gain information on the isomorphic structure of the subspaces of a twisted Hilbert space. As a sample of our results, the only spreading models of $Z(X)$ are $\ell _2$ and $\ell _M$, whenever $X$ is as above and $p\neq 2$.

Linear support for the prime number sequence and the first and second Hardy-Littlewood conjectures

2024-02-12T12:05:41+01:00

Servais and Grün used results about linear support for the prime number sequence to obtain upper bounds on the smallest prime in odd perfect numbers. This was extended by Cohen and Hendy who proved that for every $n \in \mathbb {N}$ there exists an integer $b_n$ such that $p_i \ge p_1 + ni - b_n$ for any sequence $(p_i)$ of odd primes, and found minimal values of $b_n$ for $3 \le n \le 5$, and conjectured minimal values for $6 \le n \le 10$. We give a new proof of the existence of $b_n$ and the values for $3 \le n \le 5$. We also show that if we assume that the second Hardy-Littlewood conjecture, $\pi (a+b) \le \pi (a) + \pi (b)$ for $a, b \ge 2$, is true, then we can in a finite number of steps determine numbers, $T_n$, that give quite close bounds for the values of $b_n$, namely $T_n \le b_n \le T_n + n - 2$, and determine the values of $T_n$ for $6 \le n \le 20$.

We also consider the question of whether the values of $b_n$ can be replaced by smaller numbers if we assume that $p_1 > 3$. We will show that if we assume that the first Hardy-Littlewood conjecture is true, then we can determine the minimum such values, $a_n$, for $3 \le n \le 5$. We also determine some lower bounds for $a_n$ for $n \ge 6$. It is well-known that the two Hardy-Littlewood conjectures are mutually exclusive, but we never use both conjectures simultaneously.

These results give us the upper bound $p_1<\frac {n}{2^n-1}t+b_n$ on the smallest prime in odd perfect numbers, where $t$ is the number of distinct primes dividing the odd perfect number. This bound was also found by Cohen and Hendy, but we improve the constants $b_n$.

A new invariant of lattice polytopes

2024-02-16T11:31:12+01:00

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $\omega (K[\mathcal{P}])$ of the toric ring $K[\mathcal{P}]$ defined by a lattice polytope $\mathcal{P}$ will be studied. It is shown that if $\mathcal{P}$ possesses an interior lattice point, then the maximal degree is at most $\dim \mathcal{P} - 1$, and that this bound is the best possible in general.

Special homogeneous curves

2024-01-12T10:55:17+01:00

We classify all special homogeneous curves. A special homogeneous curve $\mathcal {H}$ consists of connected components of the hyperbolic points in the level set $\{h=1\}$ of a homogeneous polynomial $h$ in two real variables of degree at least three, and admits a transitive group action of a subgroup $G\subset \mathrm {GL}(2)$ on $\mathcal {H}$ that acts via linear coordinate change.

Special folding of quivers and cluster algebras

2024-02-14T10:38:48+01:00

We give a precise definition of folded quivers and folded cluster algebras. We define a special folding of a quiver as one which cannot be associated with a skew-symmetrizable exchange matrix. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite mutation type quivers $X_6$ and $X_7$. We also construct a folded cluster algebra associated to punctured surfaces which allow for self-folded triangles. We give a simple construction of a folded cluster algebra for which the cluster complex is a generalized permutohedron.

Permanence of the torsion-freeness property for divisible discrete quantum subgroups

2024-01-30T11:23:58+01:00

We prove that torsion-freeness in the sense of Meyer-Nest is preserved under divisible discrete quantum subgroups. As a consequence, we obtain some stability results of the torsion-freeness property for relevant constructions of quantum groups (quantum (semi-)direct products, compact bicrossed products and quantum free products). We improve some stability results concerning the Baum-Connes conjecture appearing already in a previous work of the author. For instance, we show that the (resp. strong) Baum-Connes conjecture is preserved by discrete quantum subgroups (without any torsion-freeness or divisibility assumption).

Analyticity theorems for parameter-dependent plurisubharmonic functions

2024-02-13T13:43:21+01:00

In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous.

Limiting absorption principle and radiation conditions for Schrödinger operators with long-range potentials

2024-02-12T13:47:49+01:00

We show Rellich's theorem, the limiting absorption principle, and a Sommerfeld uniqueness result for a wide class of one-body Schrödinger operators with long-range potentials, extending and refining previously known results. Our general method is based on elementary commutator estimates, largely following the scheme developed recently by Ito and Skibsted.

On the shape of correlation matrices for unitaries

2023-12-28T11:40:01+01:00

For a positive integer $n$, we study the collection $\mathcal {F}_{\mathrm {fin}}(n)$ formed of all $n\times n$ matrices whose entries $a_{ij}$, $1\leq i,j\leq n$, can be written as $a_{ij}=\tau (U_j^*U_i)$ for some $n$-tuple $U_1, U_2, …, U_n$ of unitaries in a finite-dimensional von Neumann algebra $\mathcal {M}$ with tracial state τ. We show that $\mathcal {F}_{\mathrm {fin}}(n)$ is not closed for every $n\geq 8$. This improves a result by Musat and R{ø}rdam which states the same for $n\geq 11$.

Two classes of $C^$-power-norms based on Hilbert $C^$-modules

2024-01-21T13:00:42+01:00

Let $\mathfrak {A}$ be a $C^*$-algebra with the multiplier algebra $\mathcal {L}( \mathfrak {A})$. In this paper, we expand upon the concepts of “strongly type-$2$-multi-norm" introduced by Dales and “2-power-norm" introduced by Blasco, adapting them to the context of a left Hilbert $\mathfrak {A}$-module $\mathscr {E}$. We refer to these adapted notions as $\mathscr {P}_0(\mathscr {E})$ and $\mathscr {P}_2(\mathscr {E})$, respectively. Our objective is to establish key properties of these extended concepts.

We establish that a sequence of norms $(\lVert \cdot \rVert _k :k\in \mathbb {N})$ belongs to $\mathscr {P}_0(\mathscr {E})$ if and only if, for every operator $T$ in the matrix space $\mathbb {M}_{n\times m}(\mathcal {L}( \mathfrak {A}))$, the norm of $T$ as a mapping from $\ell ^2_m(\mathfrak {A} )$ to $\ell ^2_n(\mathfrak {A} )$ equals the norm of the corresponding mapping from $(\mathscr {E}^m,\lVert \cdot \rVert _m )$ to $(\mathscr {E}^n,\lVert \cdot \rVert _n )$. This characterization is a novel contribution that enriches the broader theory of power-norms. In addition, we prove the inclusion $\mathscr {P}_0(\mathscr {E})\subseteq \mathscr {P}_2(\mathscr {E})$. Furthermore, we demonstrate that for the case of $\mathfrak {A}$ itself, we have $\mathscr {P}_0(\mathfrak {A})=\mathscr {P}_2(\mathfrak {A})=\lbrace ( \lVert \cdot \rVert _{\ell ^2_k(\mathfrak {A} ) } :k\in \mathbb {N} )\rbrace $. This extension of Ramsden's result shows that the only type-$2$-multi-norm based on ℂ is $(\lVert \cdot \rVert _{\ell ^2_k } :k\in \mathbb {N} )$. To provide concrete insights into our findings, we present several examples in the paper.

Jordan norms for multilinear maps on $C^{\ast}$-algebras and Grothendieck's inequalities

2024-02-19T14:26:42+01:00

There exists a generalization of the concept completely bounded norm, for multilinear maps on $C^{\ast }$-algebras. We will use the word Jordan norm, for this norm and denote it by $\lVert \cdot \rVert _J$. The Jordan norm $\lVert\Phi\rVert_J$ of a multilinear map is obtained via factorizations of $\Phi$ in the form $$\Phi (a_1, \dots , a_n) = T_0 \sigma _1(a_1)T_1 \cdots T_{(n-1)}\sigma _n(a_n)T_n ,$$ where the maps $\sigma _i$ are Jordan homomorphisms. We show that any bounded bilinear form on a pair of $C^{\ast }$-algebras is Jordan bounded and satisfies $\lVert B\rVert _J \leq 2\lVert B\rVert $.

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2024-05-27T10:33:59+02:00