idempotents in nest algebras abstract the algebraic equivalence and similarity classes of idempotents within a nest algebra alg are completely characterized we obtain necessary and sufficient conditions for two idempotents to be equivalent or similar our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a dimension function from into n we complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents also to each sequence of dimension functions there corresponds a commuting sequence of idempotents a criterion is obtained for when an idempotent is similar to a subidempotent of another the mapping which sends an equivalence class or idempotent to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a type i w algebra to its center valued trace we prove that almost commuting similar idempotents are homotopic this contrasts with the situation in certain c algebras using this we show that similar simultaneously diagonalizable idempotents are homotopic and in the continuous nest case every diagonal idempotent is homotopic to a core projection

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

lie isomorphisms of nest algebras on banach spaces abstract we show that a lie isomorphism between nest algebras on banach spaces can be decomposed as where is an algebraic isomorphism or a negative of an algebraic anti isomorphism and is a linear map with image in the center vanishing on each commutator

commutativity preserving linear maps and lie automorphisms of triangular matrix algebras abstract in this article we classify linear maps from the algebra t n of n upper triangular matrices into itself satisfying ab ba 0 if and only if ab ba 0 in particular we show that for n 2 any such map is either the sum of an algebra automorphism and a map into the centre or the sum of the negative of an algebra anti automorphism and a map into the centre as a consequence lie automorphisms of the algebra t n are also classified

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

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

a new invariant for finite dimensional leibniz lie algebras abstract for an n dimensional leibniz lie algebra h over a field k we introduce a new invariant a h called the universal algebra of h as a quotient of the polynomial algebra k x i j i j 1 n through an ideal generated by n 3 polynomials we prove that a h admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on h the new object a h is the key tool in answering two open problems in lie algebra theory first we prove that the automorphism group aut l b z h of h is isomorphic to the group u g a h o of all invertible group like elements of the finite dual a h o secondly for an abelian group g we show that there exists a bijection between the set of all g gradings on h and the set of all bialgebra homomorphisms a h k g based on this all g gradings on h are explicitly classified and parameterized a h is also used to prove that there exists a universal commutative hopf algebra associated to any finite dimensional leibniz algebra h

on the real polynomial bohnenblust hille inequality abstract it was recently proved by f bayart and the fourth and fifth authors that the complex polynomial bohnenblust hille inequality is subexponential here we show that for real scalars this no longer holds moreover we show among other results that if d r m stands for the real bohnenblust hille constant for m homogeneous polynomials then lim sup m d r m 1 m 2 a quite surprising result having in mind that the exact value of the bohnenblust hille constants is still a mystery

the bohr radius of the n dimensional polydisk is equivalent to sqrt frac log n n we show that the bohr radius of the polydisk mathbb d n behaves asymptotically as sqrt log n n our argument is based on a new interpolative approach to the bohnenblust hille inequalities which allows us to prove that the polynomial bohnenblust hille inequality is subexponential

commuting centralizing traces and lie triple isomorphisms on triangular algebras revisited abstract the aim of the paper is to give a description of commuting centralizing traces of bilinear maps and lie triple isomorphisms on certain triangular algebras as consequences commuting centralizing traces of bilinear maps and lie triple isomorphisms on upper triangular matrix algebras over a 2 torsion free commutative ring are completely determined

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

group gradings on the lie and jordan algebras of block triangular matrices abstract we classify up to isomorphism all gradings by an arbitrary group g on the lie algebras of zero trace upper block triangular matrices over an algebraically closed field of characteristic 0 it turns out that the support of such a grading always generates an abelian subgroup of g assuming that g is abelian our technique also works to obtain the classification of g gradings on the upper block triangular matrices as an associative algebra over any algebraically closed field these gradings were originally described by a valenti and m zaicev in 2012 assuming characteristic 0 and g finite abelian and classified up to isomorphism by a borges et al in 2018 finally still assuming that g is abelian we classify g gradings on the upper block triangular matrices as a jordan algebra over an algebraically closed field of characteristic 0 it turns out that under these assumptions the jordan case is equivalent to the lie case

elementary gradings on the lie algebra utn the algebras utn k utn k of the upper triangular matrices over a field k are of significant importance in the theory of algebras with polynomial identities group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question in this paper we consider the lie algebra utn k utn k of all upper triangular matrices of order n we study the group gradings on this algebra it turns out that the gradings on the lie algebra utn k utn k are much more intricate than those in the associative case in this paper we describe the elementary gradings on the lie algebra utn k utn k finally we study the canonical grading on utn k utn k by the cyclic group znzn of order n we produce a finite basis of the graded polynomial identities satisfied by this grading

postsmoothing for the lms algorithm and a fixed point roundoff error analysis the effect of post algorithm smoothing on digital implementations of the least mean square lms algorithm is studied an expression is derived for the mean square error mse with post algorithm pa smoothing but without finite wordlength effects it is shown that the mse can be reduced from that of lms without pa smoothing pa smoothing results in reduced convergence speed the effect of fixed point finite precision on pa smoothing is studied an expression for the mse is derived monte carlo simulation results that support the result that digital lms with pa smoothing performs better than digital lms without pa smoothing are presented

power versus performance tradeoffs for reduced resolution lms adaptive filters low power implementation of digital adaptive filters for channel equalization and interference cancelling is an impor tant aspect of wireless communications transceiver design in this paper we propose a low power design methodology based on ex ploitation of the tradeoff between high bit resolution and low power consumption we focus on the widely used finite precision lms adaptive channel equalizer for which closed form approximations can be derived for the increase in mean square error when reducing both the number of data bits and filter coefficient bits the re sulting expressions are accurate as long as the data and coefficient wordlengths are such that the weight update signal does not fall into the finite resolution deadzone of the filter we give a sufficient condition on the wordlengths that ensures that this phenomenon known as stopping or slowdown does not occur under this con dition we obtain expressions for the power optimal bit allocation factor that determines the optimal proportion of bits to allocate to the data the remainder being allocated to the coefficients numer ical studies are presented for an exponential memory isi channel with gaussian training sequence

z graded identities of the lie algebra w1 abstract let k be a field of characteristic 0 and let w 1 be the lie algebra of the derivations of the polynomial ring k t the algebra w 1 admits a natural z grading we describe the graded identities of w 1 for this grading it turns out that all these z graded identities are consequences of a collection of polynomials of degree 1 2 and 3 and that they do not admit a finite basis recall that the ordinary non graded identities of w 1 coincide with the identities of the lie algebra of the vector fields on the line and it is a long standing open problem to find a basis for these identities we hope that our paper might be a step to solving this problem

jordan isomorphisms of triangular matrix algebras over a connected commutative ring abstract let c be a 2 torsionfree commutative ring with identity 1 and let t r c r 2 be the algebra of all upper triangular r r r 2 matrices over c then c contains no idempotents except 0 and 1 if and only if every jordan isomorphism of t r c onto an arbitrary algebra over c is either an isomorphism or an anti isomorphism

new insights on the transient and steady state behavior of the quantized lms algorithm this correspondence investigates the transient and steady state behavior of the quantized lms algorithm for gaussian inputs it is shown here that the so called stopping phenomenon is really a slow down phenomenon which because of an extremely slow convergence rate looks as if the algorithm has stopped the true steady state mse is shown to be nearly independent of the number of bits in the digital word length and very nearly the steady state mse of the infinite precision lms realization these results assume that the algorithm misadjustment effects due to coefficient quantization are negligible in comparison with those due to the stopping phenomena since the true steady state is rarely achievable with a finite number of iterations determination of the step size spl mu that minimizes the residual mse must be based on a stochastic model for the transient mode of algorithm operation it is shown that the finite word length and infinite precision design cases differ only in degree and not in kind as far as the selection of spl mu is concerned

lie triple derivations of nest algebras abstract let n be a nest on a complex separable hilbert space h and n be the associated nest algebra in this paper we prove that every lie triple derivation from n into itself is of the form x xt tx h x i where t n and h is a linear mapping from n into c such that h a b c 0 for all a b c n

decoupling inequalities for polynomial chaos let x x 1 ldots x d be a sequence of independent symmetric identically distributed random vectors with independent components the main subject of this paper is the so called decoupling inequalities i e inequalities of the form begin align e phi cq x x ldots x leq e phi q x 1 x 2 ldots x d leq e phi cq x x ldots x end align where q is a symmetric multilinear form with values in a vector space f with all diagonal terms equal to zero and phi is a convex function on f

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

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

gradings and symmetries on heisenberg type algebras abstract we describe the fine group gradings on the heisenberg algebras on the heisenberg superalgebras and on the twisted heisenberg algebras we compute the weyl groups of these gradings also the results obtained respect to heisenberg superalgebras are applied to the study of heisenberg lie color algebras

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

lie isomorphisms of reflexive algebras a lie isomorphism between algebras is called trivial if where is an algebraic isomorphism or a negative of an algebraic anti isomorphism and is a linear map with image in the center vanishing on each commutator in this paper we investigate the conditions for the triviality of lie isomorphisms from reflexive algebras with completely distributive and commutative lattices cdcsl in particular we prove that a lie isomorphism between irreducible cdcsl algebras is trivial if and only if it preserves i idempotent operators the sum of an idempotent and a scalar multiple of the identity in both directions we also prove the triviality of each lie isomorphism from a cdcsl algebra onto a csl algebra which has a comparable invariant projection with rank and corank not one some examples of lie isomorphisms are presented to show the sharpness of the conditions

concentration inequalities for non lipschitz functions with bounded derivatives of higher order building on the inequalities for homogeneous tetrahedral polynomials in independent gaussian variables due to r latala we provide a concentration inequality for not necessarily lipschitz functions f mathbb r n rightarrow mathbb r with bounded derivatives of higher orders which holds when the underlying measure satisfies a family of sobolev type inequalities begin aligned vert g mathbb eg vert p le c p vert nabla g vert p end aligned such sobolev type inequalities hold e g if the underlying measure satisfies the log sobolev inequality in which case c p le c sqrt p or the poincare inequality then c p le cp our concentration estimates are expressed in terms of tensor product norms of the derivatives of f when the underlying measure is gaussian and f is a polynomial not necessarily tetrahedral or homogeneous our estimates can be reversed up to a constant depending only on the degree of the polynomial we also show that for polynomial functions analogous estimates hold for arbitrary random vectors with independent sub gaussian coordinates we apply our inequalities to general additive functionals of random vectors in particular linear eigenvalue statistics of random matrices and the problem of counting cycles of fixed length in erd s renyi random graphs obtaining new estimates optimal in a certain range of parameters

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

factoring multi sublinear maps abstract using rademacher type maximal estimates are established for k sublinear operators with values in the space of measurable functions maurey nikishin factorization implies that such operators factor through a weak type lebesgue space this extends known results for sublinear operators and improves some results for bilinear operators for example any continuous bilinear operator from a product of type 2 spaces into the space of measurable functions factors through a banach space also included are applications for multilinear translation invariant operators

bohr s phenomenon for functions on the boolean cube abstract we study the asymptotic decay of the fourier spectrum of real functions f 1 1 n r in the spirit of bohr s phenomenon from complex analysis every such function admits a canonical representation through its fourier walsh expansion f x s 1 n f s x s where x s k s x k given a class f of functions on the boolean cube 1 1 n the boolean radius of f is defined to be the largest 0 such that s f s s f for every f f we give the precise asymptotic behavior of the boolean radius of several natural subclasses of functions on finite boolean cubes as e g the class of all real functions on 1 1 n the subclass made of all homogeneous functions or certain threshold functions compared with the classical complex situation subtle differences as well as striking parallels occur

gradings on simple algebras of finitary matrices abstract we describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries over an algebraically closed field of characteristic zero

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

linear maps preserving commutativity abstract commutativity preserving maps on the real space of all real symmetric or complex self adjoint matrices are characterized related results are given for adjoint preserving maps defined on all n n matrices these results are extended to infinite dimensions in the case of invertible maps

on herstein s lie map conjectures iii abstract in the two papers on herstein s lie map conjectures i and ii solutions of all of herstein s problems concerning lie isomorphisms and derivations were given under the assumption that the algebras involved were of sufficiently high dimension in the present paper we remove this restriction thereby solving herstein s problems in full generality

graded isomorphisms on upper block triangular matrix algebras abstract we describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras as a consequence we describe the isomorphism classes of upper block triangular matrix algebras over an algebraically closed field of characteristic zero graded by a finite abelian group

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

a nonlinear analytical model for the quantized lms algorithm the power of two step size case presents a study of the quantization effects in the finite precision lms algorithm with power of two step sizes deterministic nonlinear recursions are presented for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step size spl mu the numerical solutions of these recursions are shown to agree very closely with the monte carlo simulations during all phases of the adaptation process design examples demonstrate the selection of the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance the results obtained indicate that previous models are too conservative in predicting the converged mse for a given number of bits

on z2 graded polynomial identities of the grassmann algebra abstract let f be a field of characteristic zero and let e be the grassmann algebra of an infinite dimensional f vector space l in this paper we study the z 2 graded polynomial identities of e with respect to any fixed z 2 grading such that l is an homogeneous subspace we found explicit generators for the ideal t 2 e of graded polynomial identities of e and we determine its cocharacter and codimension sequences

a roundoff error analysis of the lms adaptive algorithm the steady state output error of the least mean square lms adaptive algorithm due to the finite precision arithmetic of a digital processor is analyzed it is found to consist of three terms 1 the error due to the input data quantization 2 the error due to the rounding of the arithmetic operations in calculating the filter s output and 3 the error due to the deviation of the filter s coefficients from the values they take when infinite precision arithmetic is used the last term is of paricular interest because its mean squared value is inversely proportional to the adaptation step size both fixed and floating point arithmetics are examined and the expressions for the final mean square error are found to be similar the relation between the quantization error and the error that occurs when adaptation possibly ceases due to quantization is also investigated

graded identities and isomorphisms on algebras of upper block triangular matrices abstract let g be an abelian group and k an algebraically closed field of characteristic zero a valenti and m zaicev described the g gradings on upper block triangular matrix algebras provided that g is finite we prove that their result holds for any abelian group g any grading is isomorphic to the tensor product a b of an elementary grading a on an upper block triangular matrix algebra and a division grading b on a matrix algebra we then consider the question of whether graded identities a b where b is an algebra with a division grading determine a b up to graded isomorphism in our main result theorem 3 we reduce this question to the case of elementary gradings on upper block triangular matrix algebras which was previously studied by o m di vincenzo and e spinelli

gradings on the real forms of the albert algebra of g2 and of f4 we find all the fine group gradings on the real forms of the albert algebra and of the exceptional lie algebras g2 and f4

minimal varieties of algebras of exponential growth abstract the exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer through this scale we can now classify all minimal varieties of given exponent and of finite basic rank as a consequence we describe the corresponding t ideals of the free algebra and we compute the asymptotics of the related codimension sequences verifying in this setting some known conjectures we also show that the number of these minimal varieties is finite for any given exponent we finally point out some relations between the exponent of a variety and the gelfand kirillov dimension of the corresponding relatively free algebras of finite rank

group gradings on the lie algebra of upper triangular matrices abstract the algebras u t n of the n n upper triangular matrices over a field k are of significant importance in the theory of algebras with polynomial identities group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question we classify the group gradings on the lie algebra u t n it was proved by valenti and zaicev in 2007 that every group grading on the associative algebra u t n is isomorphic to an elementary grading the elementary gradings on u t n are also well understood see 6 it follows from our results that there are nonelementary gradings on u t n thus the gradings on the lie algebra u t n are much more intricate than those in the associative case

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

on lie gradings iii gradings of the real forms of classical lie algebras abstract maximal abelian subgroups of diagonalizable automorphisms of lie algebra so called mad groups play a crucial role in the construction of fine gradings of lie algebra our aim is to give a description of mad groups for real forms of classical lie algebras we introduce four types of matrix subgroups of g l n c called out groups ad groups out groups and ad groups for each type of these subgroups we define a relation of equivalence the problem of classifying of all non conjugate mad groups on real forms of sl n c o n c or sp n c is transformed to the problem of classifying these equivalence classes the classification of these equivalence classes is presented here

hypercontraction principle and random multilinear forms we study a banach space valued random multilinear forms in independent real random variables extensively using the concept of hypercontractive maps between lq spaces we show that multilinear forms share with linear forms a lot of properties like comparability of lq l0 and almost sure convergence

lie isomorphisms of nest algebras abstract in this paper we characterize linear maps between two nest algebras t n and t m which satisfy the property that ab ba a b b a for all a b t n in particular it is shown that such isomorphisms only exist if n is similar to m or m

a nonlinear analytical model for the quantized lms algorithm the arbitrary step size case this paper extends conditional moment techniques previously developed for the study of nonlinear versions of the lms algorithm to the study of the effects of quantizers in the finite precision case deterministic nonlinear recursions are derived for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step sizes spl mu these recursions are solved numerically and shown to be in very close agreement with the monte carlo simulations during all phases of the adaptation process a design example is presented that demonstrates how the theory can be used to select the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

gradings on the lie algebra d4 revisited abstract we classify group gradings on the simple lie algebra l of type d 4 over an algebraically closed field of characteristic different from 2 fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism for each g grading on l we also study graded l modules assuming characteristic 0

decoupling inequalities for multilinear forms in independent symmetric random variables on considere x 1 x 2 des copies independantes d une suite x x 1 x 2 de variables aleatoires symetriques independantes et m une forme multilineaire symetrique de rang s sur r n dont les composantes a i1 a is relatives a la base standard de r n satisfont a i1 i s 0

maps preserving products xy yx on von neumann algebras nnsf of china 11071201 ynsf of china 11001230 fundamental research funds for the central universities 2010121001 ynsf of fujian 2008f3013

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

the bohnenblust hille inequality for homogeneous polynomials is hypercontractive the bohnenblust hille inequality says that the 2m m 1 norm of the coefcients of an m homogeneous polynomial p on c n is bounded by kpk1 times a constant independent of n wherekk 1 denotes the supremum norm on the polydisc d n the main result of this paper is that this inequality is hypercontractive i e the constant can be taken to be c m for some c 1 combining this improved version of the bohnenblust hille inequality with other results we obtain the following the bohr radius for the polydisc d n behaves asymptotically as p logn n modulo a factor bounded away from 0 and innity

lie ring isomorphisms between nest algebras on banach spaces let n and m be nests on banach spaces x and y over the real or complex field f and let algn and algm be the associated nest algebras respectively it is shown that a map algn algm is a lie ring isomorphism i e is additive lie multiplicative and bijective if and only if has the form a tat 1 h a i for all a algn or a ta t 1 h a i for all a algn where h is an additive functional vanishing on all commutators and t is an invertible bounded linear or conjugate linear operator when dimx t is a bijective linear transformation for some field automorphism of f when dimx

lie ideals in triangular operator algebras we study lie ideals in two classes of triangular operator algebras nest algebras and triangular uhf algebras our main results show that if 2 is a closed lie ideal of the triangular operator algebra a then there exist a closed associative ideal ic and a closed subalgebra or of the diagonal a ni a so that k c 2 c ic oc introduction let a be an associative complex algebra under the lie multiplication x y xy yx a becomes a lie algebra a lie ideal in a is a linear manifold z in a for which a k c for every a e a and k c 2 in many instances there is a close connection between the lie ideal structure and the associative ideal structure of a this connection has been investigated for prime rings in 6 in 3 for b 5 the set of bounded operators on a hilbert space si and in 10 for certain von neumann algebras see also 9 4 11 in this paper we pursue this line of investigation for two classes of triangular operator algebras namely nest algebras anld triangular uhf algebras the authors would like to thank frank a zorzitto for many helpful conversations 1 weakly closed lie ideals in nest algebras recall that a nest jk on a hilbert space si is a chain of closed subspaces of si which is closed under the operations of arbitrary intersections and closed linear spans and which includes o and s5 the nest algebra t kj is the algebra of all operators on se leaving every member of jk invariant this is always closed in the weak operator topology the diagonal d kj of a nest algebra t jk is the von neumann subalgebra 7 k n t k if e f e kv with e f then f e is called an interval of the nest the nonzero minimal intervals are called atoms a nest is atomic if the atoms of the nest span 55 we refer the reader to 1 for more information on nest algebras our main result theorem 12 shows that for every weakly closed lie ideal z in 7 jk there exist a corresponding weakly closed associative ideal ic and a von neumann subalgebra sk of o a such that

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

on the fourier spectrum of functions on boolean cubes let f be a real valued function of degree d defined on the n dimensional boolean cube pm 1 n and f x sum s subset 1 ldots n widehat f s prod k in s x k its fourier walsh expansion the main result states that there is an absolute constant c 0 such that the ell 2d d 1 sum of the fourier coefficients of f pm 1 n longrightarrow 1 1 is bounded by c sqrt d log d it was recently proved that a similar result holds for complex valued polynomials on the n dimensional polytorus mathbb t n but that in contrast to this a replacement of the n dimensional torus mathbb t n by the n dimensional cube 1 1 n leads to a substantially weaker estimate this in the boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case we indicate how our result is linked with several questions in quantum information theory

the symmetries of the fine gradings of sl nk c associated with direct product of pauli groups a grading of a lie algebra is called fine if it could not be further refined for a fine grading of a simple lie algebra we define its weyl group to describe the symmetry of this grading it is already known that the weyl group of the fine grading of sl n c induced by the action of the group n of the generalized pauli matrices of rank n is sl 2 zn where zn is the cyclic group of order n in this paper we consider the fine grading of sl nk c induced by the action of the group of k fold tensor product of the generalized pauli matrices of rank n we prove that its weyl group is sp 2k zn and is generated by transvections therefore this generalizes the previous result

graded polynomial identities and specht property of the lie algebra sl2 abstract let g be a group the lie algebra sl 2 of 2 2 traceless matrices over a field k can be endowed up to isomorphism with three distinct non trivial g gradings induced by the groups z 2 z 2 z 2 and z it has been recently shown koshlukov 2008 8 that for each grading the ideal of g graded identities has a finite basis in this paper we prove that when char k 0 the algebra sl 2 endowed with each of the above three gradings has an ideal of graded identities id g sl 2 satisfying the specht property i e every ideal of graded identities containing id g sl 2 is finitely based

decoupling of banach valued multilinear forms in independent symmetric banach valued random variables let e be a banach space and e be symmetric continuous and convex let ui and ri be independent sequences of random variables having respectively u 0 1 and symmetric bernoulli distributions and let u i j and r i j for j 1 2 d be independent copies of these sequences we prove two sided inequalities between the quantities e phi sum limits underset raise0 3em hbox smash scriptscriptstyle i in mathbb z d r i 1 r i d f underset raise0 3em hbox smash scriptscriptstyle i u i 1 u i d and their decoupled versions e phi sum limits underset raise0 3em hbox smash scriptscriptstyle iota in mathbb z d r i 1 1 r i d d f underset raise0 3em hbox smash scriptscriptstyle iota u i 1 1 u i d d for bochner integrable f i 0 1 d e this generalizes results of kwapien and of zinn

nonlinear mappings preserving product xy yx on factor von neumann algebras abstract let a and b be two factor von neumann algebras for a b a define by a b ab ba the new product of a and b in this article it is proved that a not necessarily linear bijective mapping a b satisfies a b a b for all a b a if and only if is a ring isomorphism in particular if the von neumann algebras a and b are type i factors then is a unitary isomorphism or a conjugate unitary isomorphism

gradings on the algebra of upper triangular matrices and their graded identities abstract let k be an infinite field and let ut n k denote the algebra of n n upper triangular matrices over k we describe all elementary gradings on this algebra further we describe the generators of the ideals of graded polynomial identities of ut n k and we produce linear bases of the corresponding relatively free graded algebras we prove that one can distinguish the elementary gradings by their graded identities we describe bases of the graded polynomial identities in several typical cases although in these cases we consider elementary gradings by cyclic groups the same methods serve for elementary gradings by any finite group

simple graded division algebras over the field of real numbers abstract we classify up to equivalence all finite dimensional simple graded division algebras over the field of real numbers the grading group is any finite abelian group

malliavin calculus and decoupling inequalities in banach spaces abstract we develop a theory of malliavin calculus for banach space valued random variables using radonifying operators instead of symmetric tensor products we extend the wiener ito isometry to banach spaces in the white noise case we obtain two sided l p estimates for multiple stochastic integrals in arbitrary banach spaces it is shown that the malliavin derivative is bounded on vector valued wiener ito chaoses our main tools are decoupling inequalities for vector valued random variables in the opposite direction we use meyer s inequalities to give a new proof of a decoupling result for gaussian chaoses in umd banach spaces

classification of division gradings on finite dimensional simple real algebras abstract we classify up to isomorphism and up to equivalence division gradings by abelian groups on finite dimensional simple real algebras gradings on finite dimensional simple algebras are determined by division gradings so our results give the classification up to isomorphism of not necessarily division gradings on such algebras linear algebra over the field of two elements plays an interesting role in the proofs

nest algebras and similarity transformations in recent years the theory of algebras of operators on hilbert space has been stimulated by developments in the theory of quasitriangularity andersen 1 has shown that up to unitary equivalence there is only one continuous quasitrianguilar algebra we use this to provide the following answer to a question posed by j r ringrose approximately 20 years ago similar continuous nests on separable hilbert space can fail to be unitarily equivalent theorem 2 2 a consequence is the existence of a nonhyperintransitive compact operator corollary 2 3 which answers a question of kadison and singer 12 and of gohberg and krein 11 we extend our initial theorem to show that arbitrary continuous nests are similar theorem 2 10 and that every maximal nest is similar to a multiplicity one nest theorem 2 11 a consequence is that every compact operator is similar to a hyperintransitive compact operator corollary 2 12 the similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator the methods of section 2 apply only to the continuous parts of nests for general results the atomic core part must be dealt with in section 3 different methods are developed for this purpose again utilizing andersen s results these are used in section 4 to prove that a complete nest x admits an arveson factorization for every positive invertible operator if and only if x is countable as a family of subspaces theorem 4 7 a consequence is that if x is an uncountable nest with atomic core then some similarity transformation of x has a continuous part this could not be deduced from section 2 these methods also yield a weak factorization result which concludes the paper it should be noted that a negative resolution to the ringrose question was conjectured in recent years by several mathematicians including w arveson and j ringrose also many of the results presented in this paper were announced in an a m s bulletin article 16 finally we wish to thank the referee for suggesting that the original manuscript could be condensed and improved

local lie derivations of nest algebras abstract let n be a nest on a hilbert space h and alg n be the associated nest algebra we prove that every local lie derivation from alg n into b h is a lie derivation

specht property for some varieties of jordan algebras of almost polynomial growth abstract let f be a field of characteristic zero in 25 it was proved that u j 2 the jordan algebra of 2 2 upper triangular matrices can be endowed up to isomorphism with either the trivial grading or three distinct non trivial z 2 gradings or by a z 2 z 2 grading in this paper we prove that the variety of jordan algebras generated by u j 2 endowed with any g grading has the specht property i e every t g ideal containing the graded identities of u j 2 is finitely based moreover we prove an analogue result about the ordinary identities of a 1 a suitable infinitely generated metabelian jordan algebra defined in 27

graded polynomial identities for the upper triangular matrix algebra over a finite field abstract let k be a finite field and let u t n k be the algebra of n n upper triangular matrices over k in this paper we describe the set of all g graded polynomial identities of u t n k where g is any group moreover we describe a linear basis for the corresponding relatively free graded algebra

automorphisms of the lie algebra of upper triangular matrices over a connected commutative ring abstract let r be a non trivial commutative ring having no idempotents except 0 and 1 denote by t the lie algebra over r consisting of all upper triangular n by n matrices over r we give an explicit description of the automorphism group of this lie algebra

group gradings on the jordan algebra of upper triangular matrices abstract let g be an arbitrary group and let k be a field of characteristic different from 2 we classify the g gradings on the jordan algebra uj n of upper triangular matrices of order n over k it turns out that there are up to a graded isomorphism two families of gradings the elementary gradings analogous to the ones in the associative case and the so called mirror type mt gradings moreover we prove that the g gradings on uj n are uniquely determined up to a graded isomorphism by the graded identities they satisfy

gradings on the real form 6 26 we describe four fine gradings on the real form x1d522 6 26 of the complex lie algebra x1d522 6 they are precisely the gradings whose complexifications are fine gradings on the complex algebra the universal grading groups are z26 z z24 z2 z23 and z4 z24

gradings on the simple real lie algebras of types g2 and d4 abstract we classify group gradings on the simple lie algebras of types g 2 and d 4 over the field of real numbers or any real closed field fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism