group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

solutions to operator equations on hilbert c modules abstract let a be a c algebra e f and g be hilbert a modules t l a e f and t l a g f we generalize the douglas theorem about the operator equation tx t from hilbert space to hilbert c module to the equation tx t and to the system of two equations tx t and xs s we get the forms of general solutions in the case that there exists a solution and give some sufficient and necessary conditions for the existence of solutions and the existence of hermitian solutions and positive solutions in the case g e in addition the forms of general hermitian solution and general positive solution in the case that there exists a solution and g e to the equation tx t are given too

new extremal characterizations of generalized inverses of linear operators abstract in 1956 r penrose studied best approximate solutions of the matrix equation ax b he proved that a b where a is the moore penrose inverse is the unique matrix of minimal frobenius norm among all matrices which minimize the frobenius norm of ax b in particular a is the unique best approximate solution of ax i the vector version of penrose s result that is the fact that the vector a b is the best approximate solution in the euclidean norm of the vector equation ax b has long been generalized to infinite dimensional hilbert spaces in this paper an infinite dimensional version of penrose s full result is given we show that a straightforward generalization is not possible and provide new extremal characterizations in terms of the hermitian order of a and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least squares solutions for a certain class of operators we can phrase our characterizations in terms of a whole class of norms including the hilbert schmidt and the trace norms thus providing new extremal characterizations even in the matrix case we treat both operators with closed range and with not necessarily closed range finally we characterize a as the unique inner inverse of minimal hilbert schmidt norm if a 2

reduced solutions of douglas equations and angles between subspaces in this paper we study some particular solutions of douglas type equations by means of generalized inverses and angles we apply this result to characterize positive solutions and some special projections which are symmetrizable for a semidefinited positive bounded linear operator

extrinsic radius pinching in space forms of nonnegative sectional curvature we first give new estimates for the extrinsic radius of compact hypersurfaces of the euclidean space mathbb r n 1 and the open hemisphere in terms of high order mean curvatures then we prove pinching results corresponding to these estimates we show that under a suitable pinching condition m is diffeomorphic and almost isometric to an n dimensional sphere

joint extensions in families of contractive commuting operator tuples abstract in this paper we systematically study extension questions in families of commuting operator tuples that are associated with the unit ball in c d

a pinching theorem for homotopy spheres the length excess of a triangle measures how much the triangle inequality fails to be an equality this notion was first studied seriously in ag we say that a bounded metric space x x d has excess 5 e if there are points p q e x such that d p x d x q d p q e for all x ex the excess e x of x is the infimum of all e 0 where x has excess e notice that the standard spheres have excess o in general a closed riemannian manifold m has excess 0 if and only if there is a point p e m whose cutlocus c p is a single point q em such a manifold is clearly a twisted sphere i e it is the union of two discs glued together along their common boundary conversely according to a result of weinstein b appendix c 4 any twisted sphere carries a riemannian metric for which there is a point whose cut locus is another point with this in mind we are interested in closed riemannian manifolds with small excess however without any further geometric restrictions small excess has no topological significance see problem 8 though in fact any closed n manifold m can be given a riemannian metric withdiam m 1c and arbitrarily small excess simply make sure that all topological complexities of m i e the complement of a topological n disc are contained in a sufficiently small metric ball and what remains is metrically close to the complement of a small ball in the unit sphere sn 1 our main result is theorem a given k e r d v 0 and an integer n 2 there is an t e n k d v such that any closed riemannian n mani old m with sectional curvature km k diamm d volm v and e m e is a homotopy sphere

factorizations of linear relations abstract given two linear relations a and b we characterize the existence of a linear relation operator c such that a b c respectively a c b these factorizations extend and improve well known results by douglas and sebestyen

classification of involutions on graded division simple real algebras abstract we classify up to isomorphism and up to equivalence involutions on graded division finite dimensional simple real associative algebras when the grading group is abelian

partial isometries in semi hilbertian spaces in this work the concepts of isometry unitary and partial isometry on a hilbert space are generalized when an additional semi inner product is considered these new concepts are described by means of oblique projections

gradings on classical central simple real lie algebras abstract for any abelian group g we classify up to isomorphism all g gradings on the classical central simple lie algebras except those of type d 4 over the field of real numbers or any real closed field

on eigenvalue pinching in positive ricci curvature we shall show that for manifolds with ric n 1 the radius is close to iff the n 1 st eigenvalue is close to n this extends results of cheng and croke which show that the diameter is close to iff the first eigenvalue is close to n we shall also give a new proof of an important theorem of colding to the effect that if the radius is close to then the volume is close to that of the sphere and the manifold is gromov hausdorff close to the sphere from work of cheeger and colding these conditions imply that the manifold is diffeomorphic to a sphere

classification of group gradings on simple lie algebras of types a b c and d abstract for a given abelian group g we classify the isomorphism classes of g gradings on the simple lie algebras of types a n n 1 b n n 2 c n n 3 and d n n 4 in terms of numerical and group theoretical invariants the ground field is assumed to be algebraically closed of characteristic different from 2

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

sharp and rigid isoperimetric inequalities in metric measure spaces with lower ricci curvature bounds we prove that if x mathsf d mathfrak m is a metric measure space with mathfrak m x 1 having in a synthetic sense ricci curvature bounded from below by k 0 and dimension bounded above by n in 1 infty then the classic levy gromov isoperimetric inequality together with the recent sharpening counterparts proved in the smooth setting by milman for any k in mathbb r n ge 1 and upper diameter bounds holds i e the isoperimetric profile function of x mathsf d mathfrak m is bounded from below by the isoperimetric profile of the model space moreover if equality is attained for some volume v in 0 1 and k is strictly positive then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions finally we also establish the almost rigidity if the equality is almost attained for some volume v in 0 1 and k is strictly positive then the space must be mgh close to a spherical suspension to our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying ricci curvature lower bounds examples of spaces fitting our assumptions include measured gromov hausdorff limits of riemannian manifolds satisfying ricci curvature lower bounds alexandrov spaces with curvature bounded from below finsler manifolds endowed with a strongly convex norm and satisfying ricci curvature lower bounds the result seems new even in these celebrated classes of spaces

embeddings of riemannian manifolds with heat kernels and eigenfunctions we show that any closed n dimensional riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points both this time and this number can be bounded in terms of the dimension a lower bound on the ricci curvature the injectivity radius and the volume it follows that the manifold can be embedded by a finite number of eigenfunctions of the laplace operator again this number only depends on the geometric bounds and the dimension in addition both maps can be made arbitrarily close to an isometry in the appendix we derive quantitative estimates of the harmonic radius so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well 2016 wiley periodicals inc

an eigenvalue pinching theorem in 1958 lichnerowicz 7 showed that for a compact n dimensional riemannian manifold m whose ricci curvature is bounded below by n 1 the first non zero eigenvalue 21 of the laplacian satisfies 2 t n if in fact 21 n obata proved that m must be isometric to the standard sphere a natural question is do there exist constants c n 1 depending only on n such that if m is as above and c n n 21 n then m must be diffeomorphic to a sphere here by combining the works of gromov 3 berard and meyer 1 and grove and shiohama 4 we show

graded involutions on block triangular matrix algebras abstract in this paper we classify the ordinary and graded involutions on block triangular matrix algebras over an algebraically closed field of characteristic 2

group gradings on superinvolution simple superalgebras in this paper we describe all group gradings by an arbitrary finite group g on non simple finite dimensional superinvolution simple associative superalgebras over an algebraically closed field f of characteristic 0 or coprime to the order of g

manifolds of positive ricci curvature with almost maximal volume 10 in this note we consider complete riemannian manifolds with ricci curvature bounded from below the well known theorems of myers and bishop imply that a manifold mn with ric n 1 satisfies diam mn diam sn l vol mn vol sn l it follows from ch that equality in either of these estimates can be achieved only if mn is isometric to sn 1 the natural conjecture is that a manifold mn with almost maximal diameter or volume must be a topological equivalent to sn with respect to diameter this is true only if mn satisfies some additional assumptions see an 0 gp e with respect to volume however no extra restriction is necesary

ricci curvature bounds and einstein metrics on compact manifolds where f mo m1 is a homeomorphism and dil f is the dilatation of f given by dil f supx x2 dist f x f x2 dist x1 x2 if mo and m1 are not homeomorphic define dl mo mi oo gromov 20 proves the remarkable result that the space of compact riemannian manifolds f a 3 d of sectional curvature iki 3 0 and diameter dm v and diameter dm c iki dm vm1 in particular gromov s compactness theorem may be strengthened to the statement that f a v d is c1 l compact in the lipschitz topology in this paper we study the question of lipschitz convergence of compact riemannian manifolds with bounds imposed on the ricci curvature ric in

a note on majorization and range inclusion of adjointable operators on hilbert c modules abstract the famous douglas theorem contains four equivalent conditions on the solvability of the operator equation a x b associated to hilbert spaces which was generalized to the hilbert c module case by fang et al 2009 5 in this paper we prove that in the hilbert c module case the majorization condition and the range inclusion condition are in fact not equivalent

discrete homotopies and the fundamental group abstract we generalize and strengthen the theorem of gromov that the fundamental group of any compact riemannian manifold of diameter at most d has a set of generators g 1 g k of length at most 2 d and relators of the form g i g m g j in particular we obtain an explicit bound for the number k of generators in terms of the number of short loops at every point and the number of balls required to cover a given semi locally simply connected geodesic space as a corollary we obtain a fundamental group finiteness theorem new even for riemannian manifolds that replaces the curvature and volume conditions of anderson and the 1 systole bound of shen wei by more general geometric hypothesis implied by these conditions this theorem in turn is a special case of a theorem for arbitrary compact geodesic spaces proved using the method of discrete homotopies introduced by the first author and v n berestovskii central to the proof is the notion of homotopy critical spectrum introduced in this paper as a natural consequence of discrete homotopy methods this spectrum is closely related to the sormani wei covering spectrum which is a subset of the classical length spectrum studied by de verdiere and duistermaat guillemin it is completely determined including multiplicity by special closed geodesics called essential circles

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

gradings on simple jordan and lie algebras abstract in this paper we describe all group gradings by a finite abelian group g of several types of simple jordan and lie algebras over an algebraically closed field f of characteristic zero

examples of manifolds of positive ricci curvature on presente de nouveaux exemples de varietes de riemann simplement connexes de dimension 7 a courbure de ricci positive qui n admettent pas de metrique de courbure sectionnelle non negative dans les cas compacts et non compacts

the douglas property for multiplier algebras of operators abstract for a collection of reproducing kernels k which includes those for the hardy space of the polydisk and the ball and for the bergman space k is a complete pick kernel if and only if the multiplier algebra of h 2 k has the douglas property consequences for solving the operator equation a x y are examined

involutions on graded matrix algebras abstract in this paper we describe graded antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of characteristic different from 2

convergence and the length spectrum abstract the author defines and analyzes the 1 k length spectra l 1 k m whose union over all k n is the classical length spectrum these new length spectra are shown to converge in the sense that lim k k 1 k m i l 1 k m 0 as m i m in the gromov hausdorff sense energy methods are introduced to estimate the shortest element of l 1 k as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension a number of gap theorems are proven including one for manifolds m n with ricci n 1 and volume close to vol s n many results in this paper hold on compact length spaces in addition to riemannian manifolds

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

a sphere theorem for manifolds of positive ricci curvature instead of injectivity radius the contractibility radius is estimated for a class of complete manifolds such that ricm 1 km k 2 and the volume of m the volume of the r e ball on the unit m sphere m dim m then for a suitable choice of e e m k every m belonging to this class is homeomorphic to

large manifolds with positive ricci curvature the main purpose of this paper is to show that an n dimensional manifold with ricci curvature greater or equal to n 1 which is close in the gromov hausdor topology to the unit n sphere has volume close to that of the sphere this shows the converse of the theorem in c1 namely together with c1 it shows that an n manifold with ricci curvature greater or equal to n 1 is close to the sphere if and only if the volume is close to that of the sphere in particular by p such a manifold is homeomorphic to a sphere further as an application of this and the result of c1 we prove a radius theorem saying that if an n manifold with ricci curvature greater or equal to n 1 has radius almost equal to then the volume is close to that of the sphere in order to obtain these results we further develop and apply the estimates of c1 whereas the main concern in c1 were with the large scale geometry the main concern of this paper is with the small scale geometry let n be the volume of the round n sphere sn with sectional curvature one

alexandrov spaces with large volume growth abstract let x d be an n dimensional alexandrov space whose hausdorff measure h n satisfies a condition giving the metric measure space x d h n a notion of having nonnegative ricci curvature we examine the influence of large volume growth on these spaces and generalize some classical arguments from riemannian geometry showing that when the volume growth is sufficiently large then x d h n has finite topological type

hausdorff convergence and universal covers we prove that if y is the gromov hausdorff limit of a sequence of compact manifolds m n i with a uniform lower bound on ricci curvature and a uniform upper bound on diameter then y has a universal cover we then show that for i sufficiently large the fundamental group of mi has a surjective homeomorphism onto the group of deck transforms of y finally in the non collapsed case where the mi have an additional uniform lower bound on volume we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality a number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the mi are only assumed to be compact length spaces with a uniform upper bound on diameter

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

lie gradings on associative algebras abstract in this paper we apply the method of functional identities to the study of group gradings by an abelian group g on simple lie algebras under very mild restrictions on the grading group or the base field of coefficients