On a strange invariant bilinear form on the space of automorphic forms

Let F be a global field and \(G:=SL(2)\). We study the bilinear form \({{\mathcal {B}}}\) on the space of K-finite smooth compactly supported functions on \(G({\mathbb {A}})/G(F)\) defined by

$$\begin{aligned} {{\mathcal {B}}}(f_1,f_2):={{\mathcal {B}}}_{\mathrm {naive}}(f_1,f_2)-\langle M^{-1}{{\mathrm{{CT}}}}(f_1)\, ,{{\mathrm{{CT}}}}(f_2)\rangle , \end{aligned}$$

where \({{\mathcal {B}}}_{\mathrm {naive}}\) is the usual scalar product, \({{\mathrm{{CT}}}}\) is the constant term operator, and M is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and ‘geometric’ theory of automorphic forms. We also show that the form \({{\mathcal {B}}}\) is related to S. Schieder’s Picard–Lefschetz oscillators.

     

On the Topological Stable Rank of a Noncommutative Version of the Disc Algebra

Based on a bounded approximate version of LU decomposition, we calculate the topological stable rank of \({{\mathcal {A}}}_{{\mathcal {N}}}\), which is a noncommutative version of the disc algebra. In addition, a general characterization of the maximal two-sided ideals of \({{\mathcal {A}}}_{{\mathcal {N}}}\) will be obtained.     

Maximal essential extensions in the context of frames

We show that every frame can be essentially embedded in a Boolean frame, and that this embedding is the maximal essential extension of the frame in the sense that it factors uniquely through any other essential extension. This extension can be realized as the embedding \(L \rightarrow \mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\), where \(L \rightarrow \mathcal {N}(L)\) is the familiar embedding of L into its congruence frame \(\mathcal {N}(L)\), and \(\mathcal {N}(L) \rightarrow \mathcal {B}\mathcal {N}(L)\) is the Booleanization of \(\mathcal {N}(L)\). Finally, we show that for subfit frames the extension can also be realized as the embedding \(L \rightarrow {{\mathrm{S}}}_\mathfrak {c}(L)\) of L into its complete Boolean algebra \({{\mathrm{S}}}_\mathfrak {c}(L)\) of sublocales which are joins of closed sublocales.     

Invariant {\varphi}-means on left introverted subspaces with application to locally compact quantum groups

Given a Banach algebra \({{\mathcal{A}}}\), for a non-zero character \({\varphi}\) on \({{\mathcal{A}}}\), we characterize the existence of \({\varphi}\)-means on a left introverted subspace of \({{\mathcal{A}^{*}}}\) containing \({\varphi}\) in terms of certain derivations from \({{\mathcal{A}}}\) into certain Banach \({{\mathcal{A}}}\)-bimodules. We also adapt and extend a result in (Crann and Neufang, Trans Amer Math Soc 368:495–513, 2016) on locally compact quantum groups to the Banach algebra setting which, in particular, answers a question of Bédos and Tuset, concerning the amenability of locally compact quantum groups.     

Quasiorder lattices of varieties

The set \({{\mathrm{Quo}}}(\mathbf {A})\) of compatible quasiorders (reflexive and transitive relations) of an algebra \(\mathbf {A}\) forms a lattice under inclusion, and the lattice \({{\mathrm{Con}}}(\mathbf {A})\) of congruences of \(\mathbf {A}\) is a sublattice of \({{\mathrm{Quo}}}(\mathbf {A})\). We study how the shape of congruence lattices of algebras in a variety determine the shape of quasiorder lattices in the variety. In particular, we prove that a locally finite variety is congruence distributive [modular] if and only if it is quasiorder distributive [modular]. We show that the same property does not hold for meet semi-distributivity. From tame congruence theory we know that locally finite congruence meet semi-distributive varieties are characterized by having no sublattice of congruence lattices isomorphic to the lattice \(\mathbf {M}_3\). We prove that the same holds for quasiorder lattices of finite algebras in arbitrary congruence meet semi-distributive varieties, but does not hold for quasiorder lattices of infinite algebras even in the variety of semilattices.     

Complete classification of pseudo <Emphasis Type="Italic">H</Emphasis>-type Lie algebras: I

Let \({\mathscr {N}}\) be a 2-step nilpotent Lie algebra endowed with a non-degenerate scalar product \(\langle .\,,.\rangle \), and let \({\mathscr {N}}=V\oplus _{\perp }Z\), where Z is the centre of the Lie algebra and V its orthogonal complement. We study classification of the Lie algebras for which the space V arises as a representation space of the Clifford algebra \({{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\), and the representation map \(J:{{\mathrm{{\mathrm{Cl}}}}}({\mathbb {R}}^{r,s})\rightarrow {{\mathrm{End}}}(V)\) is related to the Lie algebra structure by \(\langle J_zv,w\rangle =\langle z,[v,w]\rangle \) for all \(z\in {\mathbb {R}}^{r,s}\) and \(v,w\in V\). The classification depends on parameters r and s and is completed for the Clifford modules V having minimal possible dimension, that are not necessary irreducible. We find necessary conditions for the existence of a Lie algebra isomorphism according to the range of the integer parameters \(0\le r,s<\infty \). We present a constructive proof for the isomorphism maps for isomorphic Lie algebras and determine the class of non-isomorphic Lie algebras.     

Linear sets in the projective line over the endomorphism ring of a finite field

Let \({{\mathrm{{PG}}}}(1,E)\) be the projective line over the endomorphism ring \( E={{\mathrm{End}}}_q({\mathbb F}_{q^t})\) of the \({\mathbb F}_q\)-vector space \({\mathbb F}_{q^t}\). As is well known, there is a bijection \(\varPsi :{{\mathrm{{PG}}}}(1,E)\rightarrow {\mathcal G}_{2t,t,q}\) with the Grassmannian of the \((t-1)\)-subspaces in \({{\mathrm{{PG}}}}(2t-1,q)\). In this paper along with any \({\mathbb F}_q\)-linear set L of rank t in \({{\mathrm{{PG}}}}(1,q^t)\), determined by a \((t-1)\)-dimensional subspace \(T^\varPsi \) of \({{\mathrm{{PG}}}}(2t-1,q)\), a subset \(L_T\) of \({{\mathrm{{PG}}}}(1,E)\) is investigated. Some properties of linear sets are expressed in terms of the projective line over the ring E. In particular, the attention is focused on the relationship between \(L_T\) and the set \(L'_T\), corresponding via \(\varPsi \) to a collection of pairwise skew \((t-1)\)-dimensional subspaces, with \(T\in L'_T\), each of which determine L. This leads among other things to a characterization of the linear sets of pseudoregulus type. It is proved that a scattered linear set L related to \(T\in {{\mathrm{{PG}}}}(1,E)\) is of pseudoregulus type if and only if there exists a projectivity \(\varphi \) of \({{\mathrm{{PG}}}}(1,E)\) such that \(L_T^\varphi =L'_T\).     

Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

On affine sub-families of Grain-like structures

Grain is one of eSTREAM hardware-oriented finalists. It contains a cascade connection of an 80-bit primitive linear feedback shift registers (\({{\mathrm{LFSR}}}\)) into an 80-bit nonlinear feedback shift register (\({{\mathrm{NFSR}}}\)). The variant Grain-128 has a cascade connection with both \({{\mathrm{LFSR}}}\) and \({{\mathrm{NFSR}}}\) of order 128. We consider Grain-like structures, i.e., the cascade connection of a primitive \({{\mathrm{LFSR}}}\) into an \({{\mathrm{NFSR}}}\) of the same order. It is easy to know that in such a structure, all the affine sub-families of the \({{\mathrm{NFSR}}}\) are also the affine sub-families of the cascade connection. We prove that if the degree of the characteristic function of the \({{\mathrm{NFSR}}}\) is bigger than 2, then affine sub-families of the cascade connection must also be affine sub-families of the \({{\mathrm{NFSR}}}\). The same result holds if the order of the primitive \({{\mathrm{LFSR}}}\) is bigger than the order of the \({{\mathrm{NFSR}}}\).     

Resilience of ranks of higher inclusion matrices

Let \(n \ge r \ge s \ge 0\) be integers and \(\mathcal {F}\) a family of r-subsets of [n]. Let \(W_{r,s}^{\mathcal {F}}\) be the higher inclusion matrix of the subsets in \({{\mathcal {F}}}\) vs. the s-subsets of [n]. When \(\mathcal {F}\) consists of all r-subsets of [n], we shall simply write \(W_{r,s}\) in place of \(W_{r,s}^{\mathcal {F}}\). In this paper we prove that the rank of the higher inclusion matrix \(W_{r,s}\) over an arbitrary field K is resilient. That is, if the size of \(\mathcal {F}\) is “close” to \({n \atopwithdelims ()r}\) then \({{\mathrm{rank}}}_{K}( W_{r,s}^{\mathcal {F}}) = {{\mathrm{rank}}}_{K}(W_{r,s})\), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over \({\mathbb {F}}_q\) is also resilient if \(\mathrm{char}(K)\) is coprime to q.     

A decomposition theorem for Lie ideals in nest algebras

Let \({\mathcal {N}}\) be a nest and let \({\mathcal {L}}\) be a weakly closed Lie ideal of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We explicitly construct the greatest weakly closed associative ideal \({\mathcal {J} (\mathcal {L})}\) contained in \({\mathcal {L}}\) and show that \({\mathcal {J} (\mathcal {L}) \subseteq \mathcal {L} \subseteq \mathcal {J} (\mathcal {L})\oplus {\breve{\mathcal{D}}} (\mathcal {L})}\) , where \({{\breve{\mathcal{D}}}} (\mathcal {L})\) is an appropriate subalgebra of the diagonal \({\mathcal {D} (\mathcal {N})}\) of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We show that norm-preserving linear extensions of elements of the dual of \({\mathcal {L}}\) , satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal \({\mathcal {J} (\mathcal {L})}\) .     

An Easton like theorem in the presence of Shelah cardinals

We show that Shelah cardinals are preserved under the canonical \({{\mathrm{GCH}}}\) forcing notion. We also show that if \({{\mathrm{GCH}}}\) holds and \(F:{{\mathrm{REG}}}\rightarrow {{\mathrm{CARD}}}\) is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies \(\forall \kappa \in {{\mathrm{REG}}},~ 2^{\kappa }=F(\kappa )\). This gives a partial answer to a question asked by Cody (Arch Math Logic 52(5–6):569–591, 2013) and independently by Honzik (Acta Univ Carol 1:55–72, 2015). We also prove an indestructibility result for Shelah cardinals.     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).     

On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

A property of conditional expectation

Let \(({\mathcal {M}},\tau )\) be a semi-finite von Neumann algebra, \(({\mathcal {N}},\tau |_{{\mathcal {N}}})\) be a semi-finite von Neumann subalgebra and \({\mathcal {E}}:\;{\mathcal {M}}\rightarrow {\mathcal {N}}\) be a conditional expectation which leaves \(\tau \) invariant. We proved super-majorization for the conditional expectation \({\mathcal {E}}\) and related inequalities.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Uncountable critical points for congruence lattices

The critical point between two classes \({{\mathcal K}}\) and \({{\mathcal L}}\) of algebras is the cardinality of the smallest semilattice isomorphic to the semilattice of compact congruences of some algebra in \({{\mathcal K}}\), but not in \({{\mathcal L}}\). Our paper is devoted to the problem of determining the critical point between two finitely generated congruence-distributive varieties. For a homomorphism \({\varphi: S \rightarrow T}\) of \({(0, \vee)}\)-semilattices and an automorphism \({\tau}\) of T, we introduce the concept of a \({\tau}\)-symmetric lifting of \({\varphi}\). We use it to prove a criterion which ensures that the critical point between two finitely generated congruence-distributive varieties is less or equal to \({\aleph_{1}}\). We illustrate the criterion by constructing two new examples with the critical point exactly \({\aleph_{1}}\).     

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 \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.