Real operator algebras and real completely isometric theory

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485–496, 2003, Illinois J Math 47(4):1047–1062, 2003), of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, $C^*$ -envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided $M$ -ideal theory and characterize one-sided $M$ -ideals in real $C^*$ -algebras and real operator algebras with contractive approximate identity.     

A generalization of a theorem of Imai and its applications to Iwasawa theory

It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite $K(K^{1/p^\infty })$ -rational torsion subgroup. This can be used to remove certain conditions assumed in some theorems in Iwasawa theory.     

Finding disjoint surfaces in 3-manifolds

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no two-spheres. We investigate the existence of two properly embedded disjoint surfaces $S_{1}$ and $S_{2}$ such that $M - (S_{1} \cup S_{2})$ is connected. We show that there exist two such surfaces if and only if M is neither a $\mathbb Z _{2}$ homology solid torus nor a $\mathbb Z _{2}$ homology cobordism between two tori. In particular, the exterior of a link with at least three components always contains two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

To solve variational indefinite problems, one uses classically the Banach–Ne?as–Babu?ka theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $\text{ div}\sigma \nabla u=f$ set in $H^1_0$ , with $\sigma $ exhibiting a sign change.     

Some consequences of Rado’s selection lemma

We prove in set theory without the Axiom of Choice, that Rado’s selection lemma ( ${\mathbf{RL}}$ ) implies the Hahn-Banach axiom. We also prove that ${\mathbf{RL}}$ is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, ${\mathbf{RL}}$ implies that every filter on a well orderable set is included in a ultrafilter. In set theory with atoms, the “Multiple Choice” axiom implies ${\mathbf{RL}}$ .     

Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters \alpha =\beta =-1

In this paper, we consider the second-order differential expression $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $\alpha =\beta =-1$ in contrast to the classical case when $\alpha ,\beta >-1$ . For fixed $k\ge 0$ and appropriate values of the spectral parameter $\lambda ,$ the equation $\ell [y]=\lambda y$ has, as in the classical case, a sequence of (Jacobi) polynomial solutions $\{P_{n}^{(-1,-1)} \}_{n=0}^{\infty }.$ These Jacobi polynomial solutions of degree $\ge 2$ form a complete orthogonal set in the Hilbert space $L^{2}((-1,1);(1-x^{2})^{-1})$ . Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first-degree solution, the set of polynomial solutions of degree $\ge 0$ are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator $T$ , generated by $\ell [\cdot ],$ in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman–Krein–Naimark theory is essential in helping to construct $T$ in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.     

Character sheaves on unipotent groups in positive characteristic: foundations

In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic $p>0$ . In particular, we show that every admissible pair for such a group $G$ gives rise to an $\mathbb{L }$ -packet of character sheaves on $G$ and that conversely, every $\mathbb{L }$ -packet of character sheaves on $G$ arises from a (nonunique) admissible pair. In the Appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first Appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck–Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third Appendix proves that the “naive” definition of the equivariant $\ell $ -adic derived category with respect to a unipotent algebraic group is equivalent to the “correct” one.     

Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in {\mathbb {R}}^2

The linear stability of steady-state periodic patterns of localized spots in \({\mathbb {R}}^2\) for the two-component Gierer–Meinhardt (GM) and Schnakenberg reaction–diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient \({\displaystyle \varepsilon }^2\) of the activator concentration. In the limit \({\displaystyle \varepsilon }\rightarrow 0\) , localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area \(|\Omega |\) . To leading order in \(\nu ={-1/\log {\displaystyle \varepsilon }}\) , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies \(D={D_0/\nu }\) for some \(D_0\) independent of the lattice and the Bloch wavevector \({\pmb k}\) . From a combination of the method of matched asymptotic expansions, Floquet–Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an \({\mathcal O}(\nu )\) neighborhood of the origin in the spectral plane is derived when \(D={D_0/\nu } + D_1\) , where \(D_1={\mathcal O}(1)\) is a detuning parameter. The periodic pattern is linearly stable when \(D_1\) is chosen small enough so that this continuous band is in the stable left half-plane \(\text{ Re }(\lambda )<0\) for all \({\pmb k}\) . Moreover, for both the Schnakenberg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green’s function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of \(D\) . From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green’s function that defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in \(D\) .     

Quasi-Banach operator ideals with a very strange trace

As shown by S. Lord, F. Sukochev, and D. Zanin (see [7]), the theory of singular traces is well understood for operators on the Hilbert space. The situation turns out to be completely different in the Banach space setting. Indeed, quite strange phenomena may occur. We will construct quasi-Banach operator ideals ${\mathfrak A}$ A with seemingly contradictory properties: On the one hand, ${\mathfrak A}$ A supports a continuous trace τ that vanishes at all finite rank operators, which means that τ is singular. On the other hand, ${\mathfrak A}$ A contains the identity map I _Z of an infinite-dimensional Banach space Z and τ (I _Z ) =  1. This implies that there exist operators ${T \in \mathfrak A (Z)}$ T ∈ A ( Z ) such that ${\tau (T^n) = 1}$ τ ( T n ) = 1 for ${n = 1,2,{\dots} \;}$ n = 1 , 2 , ? , which is impossible for singular traces in the case of a Hilbert space. As most counterexamples, the new operator ideals have no useful application. They provide, however, a deeper insight into the philosophy of traces.     

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space $X$ , $S$ and $T$ bounded linear operators from $X$ to $X$ such that $\Vert S\Vert , \Vert T\Vert <1$ and $w\in X$ , let us consider the IFS $\mathcal S _{w}=(X,f_{1},f_{2})$ , where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$ , for all $x\in X$ . On one hand we prove that if the operator $S$ is compact, then there exists a family $(K_{n})_{n\in \mathbb N }$ of compact subsets of $X$ such that $A_{\mathcal S _{w}}$ is not connected, for all $w\in X-\bigcup _{n\in \mathbb N } K_{n}$ . On the other hand we prove that if $H$ is an infinite dimensional Hilbert space, then a bounded linear operator $S:H\rightarrow H$ having the property that $\Vert S\Vert <1$ is compact provided that for every bounded linear operator $T:H\rightarrow H$ such that $\Vert T\Vert <1$ there exists a sequence $(K_{T,n})_{n}$ of compact subsets of $H$ such that $A_{\mathcal S _{w}}$ is not connected for all $w\in H-\bigcup _{n}K_{T,n}$ . Consequently, given an infinite dimensional Hilbert space $H$ , there exists a complete characterization of the compactness of an operator $S:H\rightarrow H$ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.     

Open Gromov–Witten invariants in dimension six

Let $L$ be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold $(X , \omega )$ . We assume that the first homology group $H_1 (L ; A)$ with coefficients in a commutative ring $A$ injects into the group $H_1 (X ; A)$ and that $X$ contains no Maslov zero pseudo-holomorphic disc with boundary on $L$ . Then, we prove that for every generic choice of a tame almost-complex structure $J$ on $X$ , every relative homology class $d \in H_2 (X , L ; \mathbb{Z })$ and adequate number of incidence conditions in $L$ or $X$ , the weighted number of $J$ -holomorphic discs with boundary on $L$ , homologous to $d$ , and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of $J$ , provided that at least one incidence condition lies in $L$ . These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring $A$ .     

Weak theories of concatenation and minimal essentially undecidable theories

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation \({\mathsf{WTC}^{-\varepsilon}}\) , which is a weak subtheory of Grzegorczyk’s theory \({\mathsf{TC}^{-\varepsilon}}\) , is a minimal essentially undecidable theory, that is, the theory \({\mathsf{WTC}^{-\varepsilon}}\) is essentially undecidable and if one omits an axiom scheme from \({\mathsf{WTC}^{-\varepsilon}}\) , then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski’s conjecture that ‘The theory \({\mathsf{TC}^{-\varepsilon}}\) is a minimal essentially undecidable theory’. For the alternative theories \({\mathsf{WTC}}\) and \({\mathsf{TC}}\) which have the empty string, we also prove that the each theory without the neutrality of \({\varepsilon}\) is to be such a theory too.     

A class of 1-generator repeated root quasi-cyclic codes

Let \(q\) be a power of a prime integer \(p, m=p^em_0\) and \(|q|_{m_{0}}\) the order of \(q\) modulo \(m_0\) . By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field \(F_q\) of length \(mn\) and index \(n\) , under the condition that \(\mathrm {gcd}(|q|_{m_0},n)=1\) , are given.     

On the commuting probability and supersolvability of finite groups

For a finite group \(G\) , let \(d(G)\) denote the probability that a randomly chosen pair of elements of \(G\) commute. We prove that if \(d(G)>1/s\) for some integer \(s>1\) and \(G\) splits over an abelian normal nontrivial subgroup \(N\) , then \(G\) has a nontrivial conjugacy class inside \(N\) of size at most \(s-1\) . We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if \(d(G)>5/16\) then either \(G\) is supersolvable, or \(G\) isoclinic to \(A_4\) , or \(G/\mathbf{Z}(G)\) is isoclinic to \(A_4\) .     

Algebraic groups over the field with one element

In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of charactéristique une, which is now called ${\mathbb{F}_1}$ , the field with one element. This idea was that every split reductive group scheme over ${\mathbb{Z}}$ should descend to ${\mathbb{F}_1}$ , and its group of ${\mathbb{F}_1}$ -rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of ${\mathbb{F}_1}$ -schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as ${\mathbb{F}_1}$ -schemes. We endow the class of ${\mathbb{F}_1}$ -schemes with two classes of morphisms, one leading to a satisfying notion of ${\mathbb{F}_1}$ -rational points, the other leading to the notion of an algebraic group over ${\mathbb{F}_1}$ such that every split reductive group is defined as an algebraic group over ${\mathbb{F}_1}$ . Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.     

Salem numbers in dynamics on Kähler threefolds and complex tori

Let \(X\) be a compact Kähler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\) . In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\) . We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\) , then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.     

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.