isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

models for n tuples of noncommuting operators abstract this paper proves versions of the rota model theorem the de branges rovnyak model theorem and the coisometric extension theorem for n tuples of not necessarily commuting operators this generalizes the work of a e frazho j funct anal 48 1982 1 11 for pairs of operators the methods involve applying the single operator results to matrices of operators

complements to models for noncommuting operators abstract this paper develops a dilation theory for two noncommutative operators many of the results and techniques in dilation theory for one operator are extended to this setting this is done by extending the schaffer construction for one operator to the fock space setting a new wold decomposition for two isometrics with orthogonal range is obtained this wold decomposition and the schaffer construction is used to extend the sz nagy foias lifting theorem and the characteristic function theory to the noncommutative setting the characteristic function is operator valued and defined in a fock space

characteristic function for polynomially contractive commuting tuples abstract in this note we develop the theory of characteristic function as an invariant for n tuples of operators the operator tuple has a certain contractivity condition put on it this condition and the class of domains in c n that we consider are intimately related a typical example of such a domain is the open euclidean unit ball given a polynomial p in c z 1 z 2 z n whose constant term is zero all the coefficients are nonnegative and the coefficients of the linear terms are nonzero one can naturally associate a reinhardt domain with it which we call the p ball definition 1 1 using the reproducing kernel hilbert space h p c associated with this reinhardt domain in c n s pott constructed the dilation for a polynomially contractive commuting tuple definition 1 2 s pott standard models under polynomial positivity conditions j operator theory 41 1999 365 389 mr 2000j 47019 given any polynomially contractive commuting tuple t we define its characteristic function t which is a multiplier we construct a functional model using the characteristic function exploiting the model we show that the characteristic function is a complete unitary invariant when the tuple is pure the characteristic function gives newer and simpler proofs of a couple of known results one of them is the invariance of the curvature invariant and the other is a beurling theorem for the canonical operator tuple on h p c it is natural to study the boundary behaviour of t in the case when the domain is the euclidean unit ball we do that and here essential differences with the single operator situation are brought out

multivariable bohr inequalities operator valued multivariable bohr type inequalities are obtained for i a class of noncommutative holomorphic functions on the open unit ball of generalizing the analytic functions on the open unit disc ii the noncommutative disc algebra and the noncommutative analytic toeplitz algebra iii a class of noncommutative selfadjoint harmonic functions on the open unit ball of generalizing the real valued harmonic functions on the open unit disc iv the cuntz toeplitz algebra the reduced resp full group algebra resp of the free group with generators v a class of analytic functions on the open unit ball of the classical bohr inequality is shown to be a consequence of fejer s inequality for the coefficients of positive trigonometric polynomials and haager up de la harpe inequality for nilpotent operators moreover we provide an inequality which for analytic polynomials on the open unit disc is sharper than bohr s inequality

irreducible finite dimensional representations of equivariant map algebras suppose a finite group acts on a scheme x and a finite dimensional lie algebra g the corresponding equivariant map algebra is the lie algebra m of equivariant regular maps from x to g we classify the irreducible finite dimensional representations of these algebras in particu lar we show that all such representations are tensor products of evaluation representations and one dimensional representations and we establish conditions ensuring that they are all evaluation representations for example this is always the case if m is perfect our results can be applied to multiloop algebras current algebras the onsager algebra and the tetrahedron algebra doing so we easily recover the known classifications of irreducible finite dimensional representations of these algebras moreover we obtain previously unknown classifica tions of irreducible finite dimensional representations of other types of equivariant map algebras such as the generalized onsager algebra

on the local solvability and the local integrability of systems of vector fields on etudie des systemes surdetermines de champs vectoriels dans un sous ensemble ouvert de r n 1 de type l j t j j t x x j 1 m ayant des coefficients analytiques on suppose que les champs de vecteurs l j satisfont la condition de frobenius

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

an operator valued poisson kernel abstract versions of the cauchy and poisson formulas in the unit ball of c n are defined by means of appropriate operator valued kernels connections with some problems of dilation theory are then made

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

similarity problems in noncommutative polydomains abstract in this paper we consider several problems of joint similarity to tuples of bounded linear operators in noncommutative polydomains and varieties associated with sets of noncommutative polynomials we obtain analogues of classical results such as rota s model theorem for operators with spectral radius less than one sz nagy characterization of operators similar to isometries or unitary operators and the refinement obtained by foias and by de branges and rovnyak for strongly stable contractions we also provide analogues of these results in the context of joint similarity of commuting tuples of positive linear maps on the algebra of bounded linear operators on a separable hilbert space an important role in this paper is played by a class of noncommutative cones associated with positive linear maps the fourier type representation of their elements and the constrained noncommutative berezin transforms associated with these elements it is shown that there is an intimate relation between the similarity problems and the existence of positive invertible elements in these noncommutative cones and the corresponding berezin kernels

integrable representations for the twisted full toroidal lie algebras the purpose of this work is to classify irreducible integrable modules of the twisted full toroidal lie algebra with finite dimensional weight spaces and non zero central charges there are three families of such modules two of which are classified

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

equivariant map superalgebras suppose a group gamma acts on a scheme x and a lie superalgebra mathfrak g the corresponding equivariant map superalgebra is the lie superalgebra of equivariant regular maps from x to mathfrak g we classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of x is finitely generated gamma is finite abelian and acts freely on the rational points of x and mathfrak g is a basic classical lie superalgebra or mathfrak sl n n n ge 1 if gamma is trivial we show that they are all tensor products of generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on x furthermore in the case that the even part of mathfrak g is semisimple we show that all such modules are in fact tensor products of evaluation modules on the other hand if the even part of mathfrak g is not semisimple more generally if mathfrak g is of type i we introduce a natural generalization of kac modules and show that all irreducible finite dimensional modules are quotients of these as a special case our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras

classification of irreducible integrable modules for toroidal lie algebras with finite dimensional weight spaces the purpose of this paper is to classify irreducible integrable modules for multi loop algebras let v be a vector space over the complex numbers let a t 1 t n be the laurent polynomials ring in n variables let va v a and let v m v t for m m1 mn n and t t1 1 t2 2 tn n let be a simple finite dimensional lie algebra over the complex numbers then a can be given a lie algebra structure see 1 1 let d1 dn be derivations defined by di x m mix m and let d be the linear span of d1 dn then a a is a lie algebra which we call a multi loop algebra the universal central externion of a is called a toroidal lie algebra and is studied in bb bc e3 e4 em so our multiloop algebras are quotients of toroidal lie algebras by centers fix a cartan subalgebra h of let be a graded homomorphism of u ha a see section 1 for details then we can define the universal highest weight module m for a let v be the unique irreducible quotient of m we then prove in theorem 3 4 that any irreducible integrable module for a is isomorphic to v by making use of a result from chari c we associate to a map from u ha by evaluating at 1 and an irreducible module v for a derivations do not act on v and therefore it cannot be extended to a now consider v a as a

highest weight modules over pre exp polynomial lie algebras abstract in this paper we introduce pre exp polynomial lie algebras which include loop algebras virasoro like algebras and some quantum torus lie algebras we study highest weight representations of these z n 1 graded lie algebras more precisely we show that non graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite dimensional weight spaces or not and they have all finite dimensional weight spaces if and only if the highest weight is an exp polynomial highest weight we also show that non graded and graded highest weight verma modules with the same highest weight are simultaneously irreducible or not and we determine necessary and sufficient conditions for a verma module to be irreducible

representations of twisted multi loop lie algebras abstract this paper describes finite dimensional irreducible representations of twisted multi loop lie algebras these representations are given in terms of the representations of finite dimensional semisimple lie algebras