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.     

Symmetric quiver Hecke algebras and <Emphasis Type="Italic">R</Emphasis>-matrices of quantum affine algebras IV

Let \(U'_q(\mathfrak {g})\) be a twisted affine quantum group of type \(A_{N}^{(2)}\) or \(D_{N}^{(2)}\) and let \(\mathfrak {g}_{0}\) be the finite-dimensional simple Lie algebra of type \(A_{N}\) or \(D_{N}\). For a Dynkin quiver of type \(\mathfrak {g}_{0}\), we define a full subcategory \({\mathcal C}_{Q}^{(2)}\) of the category of finite-dimensional integrable \(U'_q(\mathfrak {g})\)-modules, a twisted version of the category \({\mathcal C}^{(1)}_{Q}\) introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur–Weyl duality, we construct an exact faithful KLR-type duality functor \({\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}\), where \(\mathrm{Rep}(R)\) is the category of finite-dimensional modules over the quiver Hecke algebra R of type \(\mathfrak {g}_{0}\) with nilpotent actions of the generators \(x_k\). We show that \({\mathcal F}_{Q}^{(2)}\) sends any simple object to a simple object and induces a ring isomorphism Open image in new window .     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.     

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.     

On the additivity of block designs

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.     

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.     

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.     

Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

Extremal Unital Completely Positive Maps and Their Symmetries

We consider the set \(P_1({\mathcal A},{\mathcal M})\) (respectively \(CP_1({\mathcal A},{\mathcal M})\) of unital positive (completely) maps from a \(C^*\) algebra \({\mathcal A}\) to a von-Neumann sub-algebra \({\mathcal M}\) of \({\mathcal B}({\mathcal H})\), the algebra of bounded linear operators on a Hilbert space \({\mathcal H}\). We study the extreme points of the convex set \(P_1({\mathcal A},{\mathcal M})\) (\(CP_1({\mathcal A},{\mathcal M})\)) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from \(\hat{{\mathcal A}}\) to \({\mathcal M}\), where \({\mathcal A}^{**}\) is the universal enveloping von-Neumann algebra over \({\mathcal A}\). If \({\mathcal A}={\mathcal M}\) then a (completely) positive map \(\tau \) admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if \({\mathcal M}\) is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on \({\mathcal M}\) to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role.     

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.

     

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space \({\mathcal {M}}_{g}\), of genus \(g\ge 2\) closed Riemann surfaces, is a complex orbifold of dimension \(3(g-1)\) which carries a natural real structure, i.e. it admits an anti-holomorphic involution \(\sigma \). The involution \(\sigma \) maps each point corresponding to a Riemann surface S to its complex conjugate \(\overline{S}\). The fixed point set of \(\sigma \) consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside \(\mathrm {Fix}(\sigma )\) is the locus \({\mathcal {M}}_{g}(\mathbb {R})\), the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä, and R. Silhol. The complement \(\mathrm {Fix}(\sigma )-{\mathcal {M}}_{g}(\mathbb {R})\) consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that \(\mathrm {Fix}(\sigma )\) is connected.     

Correction: An Application of Free Probability to Arithmetic Functions

In this paper, we study free probability on tensor product algebra \(\mathfrak {M} = M\,\otimes _{\mathbb {C}}\,{\mathcal {A}}\) of a \(W^{*}\)-algebra M and the algebra \({\mathcal {A}}\), consisting of all arithmetic functions equipped with the functional addition and the convolution. We study free-distributional data of certain elements of \(\mathfrak {M}\), and study freeness on \(\mathfrak {M}\), affected by fixed primes.     

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.     

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 System Coupling and Donoghue Classes of Herglotz–Nevanlinna Functions

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class \({\mathfrak {M}}_\kappa \) of Herglotz–Nevanlinna functions considered by the authors earlier, we introduce “inverse” generalized Donoghue classes \({\mathfrak {M}}_\kappa ^{-1}\) of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function \(V_\Theta (z)\) of an L-system \(\Theta \) to belong to the class \({\mathfrak {M}}_\kappa ^{-1}\) presented. In addition, we establish a connection between “geometrical” properties of two L-systems whose impedance functions belong to the classes \({\mathfrak {M}}_\kappa \) and \({\mathfrak {M}}_\kappa ^{-1}\), respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes \({\mathfrak {M}}_{\kappa _1}\)(\({\mathfrak {M}}_{\kappa _1}^{-1}\)) and \({\mathfrak {M}}_{\kappa _2}\)(\({\mathfrak {M}}_{\kappa _2}^{-1}\)), then the impedance function of the coupling falls into the class \({\mathfrak {M}}_{\kappa _1\kappa _2}\). Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class \({\mathfrak {M}}={\mathfrak {M}}_0\) is coupled with any other L-system, the impedance function of the coupling belongs to \({\mathfrak {M}}\) (the absorbtion property). Observing the result of coupling of n L-systems as n goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. All major results are illustrated by various examples.     

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.