On one class of extreme extensions of a measure

We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra \mathfrakB \mathfrak{B} of sets to a broader algebra \mathfrakA \mathfrak{A} . These sets are the set ex S _μ of all extreme extensions of the measure μ and the set H _μ of all extensions defined as l(A) = [^(m)]( h(A) ),   A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra \mathfrakB

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space

For any analytic self-map j{\varphi} of {z : |z| <  1} we give four separate conditions, each of which is necessary and sufficient for the composition operator C_j{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if C_j{C_{\varphi}} is closed-range on the Bergman space \mathbbA²{\mathbb{A}^2} , then it is closed-range on B{\mathcal{B}} , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.     

Property $${(\hbar)}$$ and cellularity of complete Boolean algebras

A complete Boolean algebra \mathbbB{\mathbb{B}}satisfies property ((^h/_2p)){(\hbar)}iff each sequence x in \mathbbB{\mathbb{B}}has a subsequence y such that the equality lim sup z _n = lim sup y _n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of property ((^h/_2p)){(\hbar)}with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}is not a theorem of ZFC and that there is no cardinal \mathfrakk{\mathfrak{k}}, definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}is a theorem of ZFC. Also, we show that the set { k: each k-cc c.B.a. has ((^h/_2p) ) }{\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}is equal to [0, \mathfrakh){[0, \mathfrak{h})}or [0, \mathfrak h]{[0, {\mathfrak h}]}and that both values are consistent, which, with the known equality {k: each c.B.a. having  ((^h/_2p) ) has the k-cc } = [\mathfrak s, ￥){{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}completes the picture.     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \mathfrakg ì \mathfrakosp(p, q|2m){\mathfrak{g}\subset\mathfrak{osp}(p, q|2m)} of Riemannian supermanifolds under the assumption that \mathfrakg{\mathfrak{g}} is a direct sum of simple Lie superalgebras of classical type and possibly of a 1-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.     

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Über die Bedingung Going up für {R \subset \hat{R}}

Let (R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with Ann_R(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with Ann_R(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to H^d_{\mathfrakq/\mathfrakp}(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals \mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}.     

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the F-jumping numbers of the test ideal t(X; D, \mathfraka^t){\tau(X; \Delta, \mathfrak{a}^t)} are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K _X  + Δ is \mathbb Q{\mathbb {Q}}-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals \mathfraka{\mathfrak{a}} is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

Revisiting the Farey AF Algebra

In a recent paper, F. Boca investigates the AF algebra \mathfrakA{{\mathfrak{A}}} associated with the Farey-Stern-Brocot sequence. We show that \mathfrakA{{\mathfrak{A}}} coincides with the AF algebra \mathfrakM₁{{\mathfrak{M_{1}}}} introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K ₀-group of \mathfrakA{\mathfrak{A}} is the lattice-ordered abelian group M₁{\mathcal{M}_{1}} of piecewise linear functions on the unit interval, each piece having integer coefficients, with the constant 1 as the distinguished order unit. Using the elementary properties of M₁{\mathcal{M}_{1}} we can give short proofs of several results in Boca’s paper. We also prove many new results: among others, \mathfrakA{{\mathfrak{A}}} is a *-subalgebra of Glimm universal algebra, tracial states of \mathfrakA{{\mathfrak{A}}} are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of \mathfrakA{{\mathfrak{A}}} are essential. We describe the automorphism group of \mathfrakA{{\mathfrak{A}}} . For every primitive ideal I of \mathfrakA{{{\mathfrak{A}}}} we compute K ₀(I) and K₀(\mathfrakA/I){{K_{0}(\mathfrak{A}/I)}}.     

Stability of unitary SK<Subscript>1</Subscript> of Azumaya algebras

For an Azumaya algebra A with center C of rank n ² and a unitary involution τ, we study the stability of the unitary SK₁ under reduction. We show that if R = C ^τ is a Henselian ring with maximal ideal \mathfrakm{\mathfrak{m}} and 2 and n are invertible in R then SK₁(A, t) @ SK₁(A/ \mathfrakm A, overline t){{{\rm SK}_1}(A, \tau) \cong {{\rm SK}_1}(A/ \mathfrak{m} A, overline \tau)}.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

On inaudible curvature properties of closed Riemannian manifolds

Following Mark Kac, it is said that a geometric property of a compact Riemannian manifold can be heard if it can be determined from the eigenvalue spectrum of the associated Laplace operator on functions. On the contrary, D’Atri spaces, manifolds of type A{\mathcal{A}}, probabilistic commutative spaces, \mathfrakC{\mathfrak{C}}-spaces, \mathfrakTC{\mathfrak{TC}}-spaces, and \mathfrakGC{\mathfrak{GC}}-spaces have been studied by many authors as symmetric-like Riemannian manifolds. In this article, we prove that for closed Riemannian manifolds, none of the properties just mentioned can be heard. Another class of interest is the class of weakly symmetric manifolds. We consider the local version of this property and show that weak local symmetry is another inaudible property of Riemannian manifolds.     

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} of stable n-pointed complex curves of genus g. Each irreducible boundary component of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}} determines via the Pontrjagin–Thom construction a map from [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This detects many new torsion classes in the homology of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}}.