Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.     

Finite groups with small automizers of {\mathcal {F}}-subgroups

Let \({ \mathcal {F}}\) be a saturated formation and G a finite group such that \({N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}\) for every subgroup H of G. If the minimal non-\({ \mathcal {F}}\)-group is soluble, then \({G \in \mathcal {F}}\).     

An inverse spectral theorem for Kreĭn strings with a negative eigenvalue

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

$$f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.$$

A positive Borel measure τ on \({\mathbb R}\) is called a (canonical) spectral measure of the string \({\textsc S[L, \mathfrak{m}]}\) , if there exists an appropriately normalized Fourier transform of \({L^2(\mathfrak{m})}\) onto L ²(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) \({\int_{\mathbb R} \frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) . (2) Either \({{\rm supp} \tau \subseteq [0, \infty)}\) , or τ is discrete and has exactly one point mass in (?∞, 0). It is a deep result, going back to Kre?n in the 1950’s, that each measure with \({\int_{\mathbb R}\frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) and \({{\rm supp} \tau \subseteq [0, \infty)}\) is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with \({{\rm supp} \tau \not\subseteq [0, \infty)}\) is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.

     

The Representation of a C0-Semigroup of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

For a C₀-semigroup \({\{U(t)\}_{t \geq 0}}\) of linear operators in a Banach space \({{\mathfrak{B}}}\) with generator A, we describe the set of elements \({x \in {\mathfrak{B}}}\) whose orbits U(t)x can be extended to entire \({{\mathfrak{B}}}\)-valued functions of a finite order and a finite type, and establish the conditions under which this set is dense in \({{\mathfrak{B}}}\). The Hille problem of finding vectors \({x \in {\mathfrak{B}}}\) such that there exists the limit \({\lim\limits_{n \to \infty}\left(I + \frac{tA}{n}\right)^{n}x}\) is also solved in the paper. We prove that this limit exists if and only if x is an entire vector of the operator A, and if this is the case, then it coincides with U(t)x.     

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Lie Algebras of Slow Growth and Klein-Gordon PDE

We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:

1. 1)
   the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)

   $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
    
2. 2)
   the second isomorphism is for the Tzitzeica equation u_xy = e^u + e^??2u

   $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$

   where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).
    

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(e^u + e^??2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.     

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

A theorem due to Stieltjes’ states that if \({\{p_n\}_{n=0}^\infty}\) is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials \({C_{n+1}^{\lambda}}\) and \({C_{n-1}^{\lambda+t}}\), \({\lambda > -\frac 12}\), if 0 < t ≤ k + 1, and also to the zeros of \({C_{n+1}^{\lambda}}\) and \({C_{n-2}^{\lambda +k}}\) if \({k\in\{1,2,3\}}\). More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of \({C_{n}^{\lambda}}\) and the zeros of \({C_{n+1}^{\lambda}}\), \({k\in\{1,2,\dots,n-1\}}\) and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.     

A new generalization of Hermite’s reciprocity law

Given a partition \(\lambda \) of n, the Schur functor \({\mathbb {S}}_\lambda \) associates to any complex vector space V, a subspace \({\mathbb {S}}_\lambda (V)\) of \(V^{\otimes n}\). Hermite’s reciprocity law, in terms of the Schur functor, states that \({\mathbb {S}}_{(p)}\left( {\mathbb {S}}_{(q)}({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{(q)}\left( {\mathbb {S}}_{(p)}({\mathbb {C}}^2)\right) . \) We extend this identity to many other identities of the type \({\mathbb {S}}_{\lambda }\left( {\mathbb {S}}_{\delta }({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{\mu }\left( {\mathbb {S}}_{\epsilon }({\mathbb {C}}^2)\right) \).     

Diagonals of injective tensor products of Banach lattices with bases

Let E be a Banach lattice with a 1-unconditional basis \(\{e_i: i \in \mathbb {N}\}\). Denote by \(\Delta (\check{\otimes }_{n,\epsilon }E)\) (resp. \(\Delta (\check{\otimes }_{n,s,\epsilon }E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by \(\Delta (\check{\otimes }_{n,|\epsilon |}E)\) (resp. \(\Delta (\check{\otimes }_{n,s,|\epsilon |}E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal \(\{e_i\otimes \cdots \otimes e_i: i \in \mathbb {N}\}\) is a 1-unconditional basic sequence in both \(\check{\otimes }_{n,\epsilon }E\) and \(\check{\otimes }_{n,s,\epsilon }E\).     

$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Some results on the annihilators and attached primes of local cohomology modules

Let \((R, \mathfrak {m})\) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen–Macaulay with respect to an ideal \(\mathfrak {a}\) of R, then \({\text {Ann}}_R(H_{\mathfrak {a}}^{{\text {cd}}(\mathfrak {a}, M)}(M))={\text {Ann}}_RM/L={\text {Ann}}_RM\) and \({\text {Ass}}_R (R/{\text {Ann}}_RM)\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})={\text {cd}}(\mathfrak {a}, M)\},\) where L is the largest submodule of M such that \(\mathrm{cd}(\mathfrak {a}, L)< \mathrm{cd}(\mathfrak {a}, M)\). We also show that if \(H^{\dim M}_{\mathfrak {a}}(M)=0\), then \({\text {Att}}_R(H^{\dim M-1}_{\mathfrak {a}}(M))= \{\mathfrak {p}\in {\text {Supp}}(M)|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M-1\},\) and so the attached primes of \(H^{\dim M-1}_{\mathfrak {a}}(M)\) depend only on \({\text {Supp}}(M)\). Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with \(\mathrm{cd}(\mathfrak {a}, M)=\mathrm{cd}(\mathfrak {a}, R/{\text {Ann}}_RM)\), then \({\text {Att}}_R(H^{\mathrm{cd}(\mathfrak {a}, M)}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {V}}({\text {Ann}}_RM)|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\mathrm{cd}(\mathfrak {a}, M)\}.\) As a consequence of this, it is shown that if \(\dim M=\dim R\), then \({\text {Att}}_R(H^{\dim M}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M\}\).     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

A Note on Derivations on the Algebra of Operators in Hilbert <Emphasis Type="Italic">C</Emphasis>*-Modules

Let \({\mathfrak{M}}\) be a Hilbert C*-module on a C*-algebra \({\mathfrak{A}}\) and let \({End_\mathfrak{A}(\mathfrak{M})}\) be the algebra of all operators on \({\mathfrak{M}}\). In this paper, first the continuity of \({\mathfrak{A}}\)-module homomorphism derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) is investigated. We give some sufficient conditions on which every derivation on \({End_\mathfrak{A}(\mathfrak{M})}\) is inner. Next, we study approximately innerness of derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) for a σ-unital C*-algebra \({\mathfrak{A}}\) and full Hilbert \({\mathfrak{A}}\)-module \({\mathfrak{M}}\). Finally, we show that every bounded linear mapping on \({End_\mathfrak{A}(\mathfrak{M})}\) which behave like a derivation when acting on pairs of elements with unit product, is a Jordan derivation.     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).