Showing papers in "Publications of The Research Institute for Mathematical Sciences in 1996"

On the K-theory of Cuntz-Krieger algebras

Abstract: We extend the uniqueness and simplicity results of Cuntz and Krieger to the countably infinite case, under a row-finite condition on the matrix A. Then we present a new approach to calculating the K-theory of the Cuntz-Krieger algebras, using the gauge action of T, which also works when A is a countably infinite 0 − 1 matrix. This calculation uses a dual Pimsner-Voiculescu six-term exact sequence for algebras carrying an action of T. Finally, we use these new results to calculate the K-theory of the Doplicher-Roberts algebras.

Quantization of contact manifolds

Abstract: We show the existence of the stack of micro-differential modules on an arbitrary contact manifold, although we cannot expect the global existence of the ring of micro-differential operators.

Compact quantum groups of face type

Abstract: We give a generalization of the compact quantum group theory of S. Woronowicz, or rather its variant due to T. Koornwinder and M. Dijkhuizen. Our framework covers algebras of L-operators of lattice models efface type without spectral parameters and "Galois quantum group" of AFD Ilj-subfactors of index < 4.

Reduced free products of completely positive maps and entropy for free product of automorphisms

Abstract: The reduced free product of unital completely positive maps is defined. An invariant state on a C*-algebra for an automorphism (based on the free shift) is the composition of a state of a subalgebra with the reduced free product of expectations. Entropies for the reduced free product and the tensor product of an automorphism y with the free shift coincide with the entropy of y.

Irreducible modules of quantized enveloping algebras at roots of 1

Abstract: Let A be an associative algebra over a field. An interesting problem is to understand the structure of irreducible modules of A (of finite dimensions). More or less, this is equivalent to understand the structure of maximal left ideals of A (of finite codimensions). For the latter, it would be helpful if we know the generators of the maximal left ideals. In Lie theory, there are some infinite dimensional algebras associated to a semisimple Lie algebra g over C. We shall be only concerned with the following four of them. ( i ) The universal enveloping algebra U of g. (ii) The hyperalgebra Ut:= Uz ® zf , where Uz is the Kostant Z-form of U and ! is an algebraically closed field of prime characteristic. (iii) The quantized enveloping algebra U (over Q(v), v is an indeterminate) of g. (iv) The quantized hyperalgebra C75:= UQ[V t V i } ®Q[v i l 7-i]Q(£), where £eC* and UQ[VtV-l} is a Q|>, iT^-form of U [LI, Section 4.1, p. 243], and Q(£) is regarded as a Q[y, v~ ]-algebra through the Q-algebra homomorphism

Outer automorphism group of the ergodic equivalence relation generated by translations of dense subgroup of compact group on its homogeneous space

Abstract: We study the outer automorphism group OutRr of the ergodic equivalence relation Rr generated by the action of a lattice F in a semisimple Lie group on the homogeneos space of a compact group K. It is shown that OutRr is locally compact. If K is a connected simple Lie group, we prove the compactness of 0\itRr using the D. Witte's rigidity theorem. Moreover, an example of an equivalence relation without outer automorphisms is constructed.

The initial value problem for cubic semilinear Schro¨dinger equations

Abstract: We present local and global existence theorems for cubic semilinear Schrodinger equations. Our new results are the improvement of our previous ones ([2] , [3] , [4]). The idea of the proof consists of the energy and the decay estimates. These equations do not allow the classical energy estimates. To avoid this difficulty, we make strong use of S. Doi's method for linear Schrodinger type equations. Combining cubic nonlinearity and S. Doi's method, we obtain the improved results. §

Binormal deformation and bimicrolocalization

Abstract: The specialization functor is an important tool in algebraic geometry as well as in algebraic analysis. In the real case, it associates to a sheaf F on a real manifold X and to a submanifold M of X, a sheaf (i.e. an object of the derived category of sheaves) vM(F) on the normal bundle TMX which describes the "boundary values" of F along M. Its Fourier transform is the sheaf /%(F) of Sato's microlocalization of F along M.

Picard groups of hypersurfaces in toric varieties

Abstract: We study the structure of rational Picard groups of hypersurfaces of toric varieties. By using the fan structure associated to the ambient toric variety, an explicit basis of the Picard group is described by certain combinatorial data. We shall also discuss the application to Calabi-Yau spaces.

Classes of valuations closed under operations Galois-dual to Boolean sentence connectives

Abstract: Outline § 0. Introduction and Background 0.1. Closure and Generalized Closure 0.2. Galois Connexions 0.3. Combinations on Both Sides of a Galois Connexion 0.4. The Exchange Property 0.5. Two Auxiliary Notations 0.6. Tight Sets and the Exchange Property § 1. Super venience and Matching 1.1. Two Problems Suggested by the Idea of Supervenience 1.2. (FlipAround) and Some Related Conditions § 2. Closure under Operations Galois-Dual to Various Boolean Connectives 2.1. Negative Objects 2.2. Disjunctive Combinations 2.3. Conjunctive Combinations 2.4. Implicative Combinations 2.5. Some Further Cases 2.6. The Provenance of the Conditions on Generalized Closure Relations § 3. Determination by Classes of Valuations Closed under Matching 3.1. An Approach Suggested by the Preceding Discussion 3.2. An Unanswered Question and a Further Condition References

Asymptotic behavior of blowup solutions of a parabolic equation with the p -Laplacian

Abstract: We consider the blowup problem for ut— Apu+\u p~2u (x^ O , t> 0) under the Dirichlet boundary condition and p> 2. We derive sufficient conditions on blowing up of solutions. In particular, it is shown that every non-negative and non-zero solution blows up in a finite time if the domain O is large enough. Moreover, we show that every blowup solution behaves asymptotically like a self-similar solution near the blowup time. The Rayleigh type quotient introduced in Lemma A plays an important role throughout this paper.

An application of orthoisomorphisms to non-commutative L p -isometries

Abstract: We prove that if there exists an into linear isometry between non-commutative //-spaces then there exists an into Jordan * -isomorphism between underlying von Neumann algebras, as an application of Araki-Bunce-Wright's theorem concerning the characterization of orthogonality preserving positive maps between preduals. Moreover, we determine the structure of a linear non-commutative LMsometry when it is surjective and *-preserving.

Criteria of denseness of generalized eigenvectors of a class of compact nonselfadjoint (or compact resolvant) operators and applications

Abstract: In this work we establish sufficient conditions on the denseness of the generalized eigenvectors for a class of compact (or compact resolvent) non self adjoint operators. We can apply our results to operators arising in many fields, particularly in field's theory and in abstract second order differential equations. Our results generalize some results of Aimar and al [1] and [2], Macaev and Keldysh [11] and Dunford-Schwartz [9],

A discrete model of the integer quantum Hall effect

Abstract: A discrete model of the integer quantum Hall effect is analysed via its associated C*algebra. The relationship with the usual continuous models is established by viewing the observable algebras of each as both twisted group C*-algebras and twisted cross products. A Fredholm module for the discrete model is presented, and its Chern character is calculated. The discovery of the integer quantum Hall effect has prompted a wealth of theoretical speculation about the origin of the spectacular accuracy with which the Hall conductance is quantized. This paper presents a simple lattice model of the quantum Hall effect that generates much of the information arising from more complex models. This lattice model of the quantum Hall effect is often used as the discrete analogue of the Landau Hamiltonian in the physics literature, and the analysis of the model often requires restricting to rational values of the magnetic flux. It is here extended and recast to fit into the C*-algebraic framework, a development that allows (in § 3) the Hall conductance to be calculated for all real values of flux. The analysis of the expression for the conductance makes its stability with respect to small changes in magnetic field evident, for it is found to be the Chern number associated with the Fermi projection (when the latter lies in a gap of the spectrum of the discrete Hamiltonian). We display the equivalence with the formula found for rational flux in the physics literature by using an explicit representation of the algebra of observables. The Hall effect is often modelled by considering electrons moving on a plane under the influence of a perpendicular magnetic field and a periodic potential. We show in § 2 that for both this model and the discrete model mentioned above the

On localized weak precompactness in Banach spaces

Abstract: In this paper we make a study of Jf-weakly precompact sets A in Banach spaces. We give various characterizations of such sets by the effective use of the lifting theory, weak*-A*-dentability and a Jf-valued weak*-measurable function constructed in the case where A is non-^-weakly precompact. These results also can be regarded as generalizatio ns of corresponding ones on Pettis sets and weakly precompact sets.

Cohomology classification of self maps of sphere bundles over spheres

Abstract: §

Differential equations associated to the SU(2) WZNW model on elliptic curves

Abstract: We study the SU(2] WZNW model over a family of elliptic curves. Starting from the formulation developed in [13], we derive a system of differential equations which contains the Knizhmk-Zamolodchikov-Bernard equations [1][9]. Our system completely determines the AT-point functions and is regarded as a natural elliptic analogue of the system obtained in [12] for the projective line. We also calculate the system for the 1-point functions explicitly. This gives a generalization of the results in [7] for si (2, C)-characters.

Picard-Lefschetz theory for the universal coverings of complements to affine hypersurfaces

Abstract: We study the global monodromy on the middle homology group of the universal coverings of the complements to non-singular affine hypersurfaces which intersect the hyperplane at infinity transversely. This monodromy can be regarded as a deformation of the monodromy on the middle homology group of the affine hypersurfaces. We show that this representation becomes irreducible when the deformation parameter is generic. §0. Introduction Let Fdenote the vector space F(P, 6 ( d ) ) , and F the space F\\{0}. We assume that n>2 and d>3. Let P* (F) stand for the projective space F /C , and pr : J—»P*CT) the natural projection. This space P*tT) parameterizes all projective hypersurfaces of degree d in P. We fix a hyperplane at infinity Hx in P, and consider the affine space A ' = P\\Hoa. We define [/c ?„(/} to be the locus of all projective hypersurfaces of degree d which are non-singular and intersect H*, transversely, and define °iL to be the pull-back of U by the projection : i J—pr-KU) c T. For u^ r\", let fu denote the corresponding homogeneous polynomial of degree d. We put Xu ' = {fu = 0), Yu — XuHH^ Xu ~Xu\\Yu, and EU := A \\Xu. Communicated by K. Saito, December 11, 1995. 1991 Mathematics Subject Classification(s) : 14D05, 32S40 * Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060, Japan.

On algebraic #-cones in topological tensor algebras II. closed hulls and extremal rays

Abstract: The investigations on the structure of alg-# cones [F, #} in topological tensor algebras are continued, and they are aimed at the closed hulls and the extremal rays of such cones. Among others, it is proven that the elements of the closed hulls of a large class of alg-# cones with respect to some intermediate I.e. topologies are explicitely given by (infinite) sums of elements of (F, #}. Furthermore, a Krein-Milman like theorem is shown for some alg# cones, i.e., it is shown that there are enough extremal rays m {F, # } so that every element of {F, #} is an (infinite) sum of extremals of

Necessary conditions for Fredholmness of partial differential operators of irregular singular type

Abstract: The convergence of formal power series solutions of ordinary differential equations are extensively studied by many authors in connection with irregularities, (cf. [6] and [10].) In case of partial differential equations of regular singular type, several sufficient conditions are known, (cf. [4] and [1].) In the preceeding paper [9], we gave sufficient conditions for the Fredholmness of partial differential operators of irregular singular type of two independent variables in analytic and Gevrey spaces. Then we deduced the convergence of formal power series solutions from the Fredholmness of the operators. These conditions are expressed in terms of Toeplitz symbols, and they are equivalent to a Riemann-Hilbert factorization condition, (cf. [3] and (2.5), (2.6) , (2.7) which follow.) In this paper, we shall show the necessity of these sufficient conditions. More precisely, we will prove that the Riemann-Hilbert factorization conditions (2.6) and (2.7) are necessary and sufficient for the partial differential operators of irregular singular type to be Fredholm operators on certain analytic and Gevrey spaces. The proof of our theorem is based on the analysis of the main (principal) part of irregular singular type operators via Toeplitz operators on the torus T • = R/27rZ. In fact, we will show that the essential parts of these operators in studying Fredholm properties are precisely Toeplitz operators. This

Cancellation and non cancellation phenomena for infinite complexes

Abstract: Let L(I, J) be defined by the pull-back of CPT >#(Q, 2)< QSJ where {/, /} is a partition of all primes. We classify spaces (Q Z* C(f)} of loop-suspension of mapping cone of phantom map f'.L(l, J) *S for k = 0, 1, • • • , °° which have the same n-type for all n. In the category of finite CW-complexes, cancellation and non cancellation phenomena are well studied. In the category of infinite CW-complexes, the phenomena are less known. We study cancellation and non cancellation phenomena for infinite dimensional complexes by using above spaces.

On tuples of commuting compact operators

Abstract: The joint spectrum of commuting operators, as introduced by Taylor, has been shown by Cho and Takaguchi to be the set of joint eigenvalues in the case of matrices. These joint eigenvalues can be read off from their simultaneous upper triangularization. We prove here a similar result for compact operators on Banach spaces.

The Feynman representation for the Dirac propagator with a radially symmetric potential

Abstract: The dynamical group associated with the Dirac equation with a radially symmetric potential in four space-time dimensions is represented in terms of integrals with respect to operator valued set functions associated with the free Dirac operator. In coordinates in which c = fi = 1, the class of potentials treated includes Coulomb potentials — a/r with |a| < 1.

On the solvability of convolution equations in Beurling's distributions

Abstract: Let &'w be the space of Beurling's generalized distributions on En and %w the spaces of generalized distributions which has compact support We show that, for S e %w, S * &w = & w is equivalent to the following: Every generalized distribution u e %w with S * u e

The essential singularity of the solution of a ramified characteristic Cauchy problem

Abstract: J. Leray [L] and L. Carding, T. Kotake and J. Leray [G~K~L] have studied the singularities of the solution of a Cauchy problem with holomorphic data, when the initial surface includes some characteristic points. They have proved that the solution may be ramified around a hypersurface K. Y. Hamada [H] has studied another class of characteristic Cauchy problem. In his case, the solution may have an essential singularity, although the data are regular. Let Pu= v be our equation. We already know that we must allow u to be ramified or to have an essential singularity. Now that we understand this necessity, it would be desirable to allow v to be singular without introducing a larger class for u. [D] and [0-Y] are studies in this direction. They are generalizations of [L] and [G-K-L]. In the present paper, we consider a problem similar to the one in [H] . Although we impose a stronger condition on the operator P than in [H] , we assume a weaker condition on v: it is allowed to be singular. Moreover, by employing a symbol calculus like the one in [D], we can explain easily why u has an essential singularity even for a holomorphic v.