Strongly amenable involutive representations of involutive Banach algebras

Let \(\mathfrak{A }\) be a Banach \(*\) -algebra and let \(\varphi \) be a nonzero self-adjoint character on \(\mathfrak{A }\) . For a   \(*\) -representation \(\pi \) of \(\mathfrak{A }\) on a Hilbert space \(\mathcal{H }\) , we introduce and study strong \(\varphi \) -amenability of \(\pi \) in terms of certain states on the von Neumann algebra of bounded operators on \(\mathcal{H }\) . We then give some characterizations of this notion in terms of certain positive functionals on \(\mathfrak{A }\) . We finally investigate some hereditary properties of strong \(\varphi \) -amenability of Banach algebras.     

Stationary and invariant densities and disintegration kernels

Let \(G\) be a locally compact topological group, acting measurably on some Borel spaces \(S\) and \(T\) , and consider some jointly stationary random measures \(\xi \) on \(S\times T\) and \(\eta \) on \(S\) such that \(\xi (\cdot \times T)\ll \eta \) a.s. Then there exists a stationary random kernel \(\zeta \) from \(S\) to \(T\) such that \(\xi =\eta \otimes \zeta \) a.s. This follows from the existence of an invariant kernel \(\varphi \) from \(S\times {\mathcal {M}}_{S\times T}\times {\mathcal {M}}_S\) to \(T\) such that \(\mu =\nu \otimes \varphi (\cdot ,\mu ,\nu )\) whenever \(\mu (\cdot \times T)\ll \nu \) . Also included are some related results on stationary integration, absolute continuity, and ergodic decomposition.     

On canonical metrics on Cartan–Hartogs domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kähler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \) -function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.     

Invertible Composition Operators: The Product of a Composition Operator with the Adjoint of a Composition Operator

In this paper, we study the product of a composition operator \(C_{\varphi }\) with the adjoint of a composition operator \(C^{*}_{\psi }\) on the Hardy space \(H^2(\mathbb {D})\) . The order of the product gives rise to two different cases. We completely characterize when the operator \(C_{\varphi }C^{*}_{\psi }\) is invertible, isometric, and unitary and when the operator \(C^{*}_{\psi }C_{\varphi }\) is isometric and unitary. If one of the inducing maps \(\varphi \) or \(\psi \) is univalent, we completely characterize when \(C^{*}_{\psi }C_{\varphi }\) is invertible.     

The theme of a vanishing period

Consider a multivalued formal function of the type 1 $$\begin{aligned} \varphi (s) : = \sum _{j=0}^k\,c_j(s).s^{\lambda + m_j}.(\mathrm{Log}\,s)^j, \end{aligned}$$ where \(\lambda \) is a positive rational number, \(c_j\) is in \({{\mathrm{\mathbb {C}}}}[[s]]\) and \(m_j \in \mathbb {N}\) for \(j \in [0,k-1]\) . The theme associated with such a \(\varphi \) is the “minimal filtered integral equation” satisfied by \(\varphi \) , in a sense which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial of \(\varphi \) to be fixed. For a given \(\lambda \) , to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated with the logarithm of the monodromy in the geometric situation described below. Our aim is to construct some analytic invariants, for instance in the following situation, let \(f : X \rightarrow D\) be a proper holomorphic function defined on a complex manifold \(X\) with values in a disc \(D\) . We assume that the only critical value is \(0 \in D\) and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \(f^{-1}(0)\) . To a smooth \((p+1)\) -form \(\omega \) on \(X\) such that \(\mathrm{d}\omega = 0 = \mathrm{d}f \wedge \omega \) and to a vanishing \(p\) -cycle \(\gamma \) chosen in the generic fiber \(f^{-1}(s_0), s_0 \in D \setminus \{0\}\) , we associated a “vanishing period” \(F_{\gamma }(s) : = \int _{\gamma _s} \omega \big /\mathrm{d}f \) which has an asymptotic expansion at \(0\) of the form \((1)\) above, when \(\gamma \) is chosen in the spectral subspace of \(H_p(f^{-1}(s_0), {{\mathrm{\mathbb {C}}}})\) for the eigenvalue \(\mathrm{e}^{2i\pi .\lambda }\) of the monodromy of \(f\) . Here \((\gamma _s)_{s \in D^*}\) is the horizontal multivalued family of \(p\) -cycles in the fibers of \(f\) obtained from the choice of \(\gamma \) . The aim of this article was to study the module generated by such a \(\varphi \) over the algebra \(\tilde{\mathcal {A}}\) , which is the \(b\) -completion of the algebra \(\mathcal {A}\) generated by the operators \(\mathrm{a} : = \times s\) and \(\mathrm{b} : = \int _{0}^{s}\) .     

On the geometry of von Neumann algebra preduals

Let \(M\) be a von Neumann algebra and let \(M_\star \) be its (unique) predual. We study when for every \(\varphi \in M_\star \) there exists \(\psi \in M_\star \) solving the equation \(\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert \) . This is the case when \(M\) does not contain type I nor type III \(_1\) factors as direct summands and it is false at least for the unique hyperfinite type III \(_1\) factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of \(M_\star \) of length \(4\) . An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.     

On lower bounds for cohomology growth in p-adic analytic towers

Let \(p\) and \(\ell \) be two distinct prime numbers and let \(\Gamma \) be a group. We study the asymptotic behaviour of the mod- \(\ell \) Betti numbers in \(p\) -adic analytic towers of finite index subgroups. If \(\Theta \) is a finite \(\ell \) -group of automorphisms of \(\Gamma \) , our main theorem allows to lift lower bounds for the mod- \(\ell \) cohomology growth in the fixed point group \(\Gamma ^\Theta \) to lower bounds for the growth in \(\Gamma \) . We give applications to \(S\) -arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.     

Extremal geometry of a Brownian porous medium

The path \(W[0,t]\) of a Brownian motion on a \(d\) -dimensional torus \(\mathbb T ^d\) run for time \(t\) is a random compact subset of \(\mathbb T ^d\) . We study the geometric properties of the complement \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) for \(d\ge 3\) . In particular, we show that the largest regions in \(\mathbb T ^d{{\setminus }} W[0,t]\) have a linear scale \(\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}\) , where \(\kappa _d\) is the capacity of the unit ball. More specifically, we identify the sets \(E\) for which \(\mathbb T ^d{{\setminus }} W[0,t]\) contains a translate of \(\varphi _d(t)E\) , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) and the \(\varepsilon \) -cover time of \(\mathbb T ^d\) as \(\varepsilon \downarrow 0\) . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in \(\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]\) , where \(W_{\rho (t)}[0,t]\) is the Wiener sausage of radius \(\rho (t)\) , with \(\rho (t)\) chosen much smaller than \(\varphi _d(t)\) but not too small. The idea behind this choice is that \(\mathbb T ^d {{\setminus }} W[0,t]\) consists of “lakes”, whose linear size is of order \(\varphi _d(t)\) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of \(\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]\) as \(t\rightarrow \infty \) . Our results give a complete picture of the extremal geometry of \(\mathbb T ^d{{\setminus }} W[0,t]\) and of the optimal strategy for \(W[0,t]\) to realise extreme events.     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

Relatively congruence-free regular semigroups

Yu, Wang, Wu and Ye call a semigroup \(S\) \(\tau \) -congruence-free, where \(\tau \) is an equivalence relation on \(S\) , if any congruence \(\rho \) on \(S\) is either disjoint from \(\tau \) or contains \(\tau \) . A congruence-free semigroup is then just an \(\omega \) -congruence-free semigroup, where \(\omega \) is the universal relation. They determined the completely regular semigroups that are \(\tau \) -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is \(\mathrel {\mathcal {J}}\) -congruence-free if and only if it is either a semilattice or has a single nontrivial \(\mathrel {\mathcal {J}}\) -class, \(J\) , say, and either \(J\) is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for \(\mathrel {\mathcal {L}}\) and \(\mathrel {\mathcal {R}}\) . In the case of \(\mathrel {\mathcal {H}}\) , only the completely semisimple case is fully resolved, again specializing to those of Yu et al.     

On finite elements in f-algebras and in product algebras

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell $ -algebras, preferably in $f$ -algebras, and in product algebras. The additional structure of an associative multiplication leads to new questions and some new properties concerning the collections of finite, totally finite and self-majorizing elements. In some cases the order ideal of finite elements is a ring ideal as well. It turns out that a product of elements in an $f$ -algebra is a finite element if at least one factor is finite. If the multiplicative unit exists, the latter plays an important role in the investigation of finite elements. For the product of special $f$ -algebras an element is finite in the algebra if and only if its power is finite in the product algebra.     

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.     

On a result of Moeglin and Waldspurger in residual characteristic 2

Let \(F\) be a \(p\) -adic field, \(\mathbf G\) a connected reductive group over \(F\) , and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\) . A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\) , then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

A uniqueness theorem for functions in the extended Selberg class

Recently, we proved that every finite dimensional Alexandrov space is strongly locally Lipschitz contractible. In the present paper, we consider the set \(\mathcal M\) of all isometry classes of Alexandrov spaces of curvature \(\ge -1\) and of fixed dimension having upper diameter bound and lower volume bound, and prove that there exists a constant \(N\) depending on the parameters determining \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N\) strongly Lipschitz contractible balls. Also, we prove that there exists a constant \(N^\prime \) depending on \(\mathcal M\) such that every space in \(\mathcal M\) can be covered by at most \(N^\prime \) strongly Lipschitz contractible and convex regions.     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.