Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + dr –au – tb)² < 4(cr – ta)(ds – ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let w_n denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions w_n(0)= sin and pw'_n(0) = –cos and let _n = –aw_n(1)+cpw'_n(1).Define _n and _n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.     

Tensor Products of C*-Algebras Over Abelian Subalgebras

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard[7]). Suppose that B is another C()-algebra, then we form A_CB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, I_C = {ⁿ_i–1(f_i 1–1f_i)x|f_iC, xAB} is the ideal in AB generated by f1–1f|fC,and A _CB = AB/I_C. Then A_CB is an involutive algebra over C,and we shall be interested in deciding when A_CB is a pre-C*-algebra;that is, when is there a C*-norm on A_C B? There is a C*-semi-norm,which we denote by ||·||_C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||_C-min. Moreover, if A _C B has a C*-norm,then ||·||_C-min is a C*-norm on A_C B. The problem isto decide when ||·||_C-min is a norm. It was shown byBlanchard [7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||_C-min is a norm.In this paper we show that ||·||_C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences.     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

The Cauchy-Schwarz Norm Inequality for Elementary Operators in Schatten Ideals

For bounded Hilbert space operators X, A_n and B_n, n = 1, 2,..., for all p < and These inequalities involve some estimates for the norm of elementaryoperators with the range contained in the Schatten p-ideals.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

The Full Muntz Theorem in C[0, 1] and L1[0, 1]

The main result of this paper is the establishment of the ‘fullMüntz Theorem’ in C[0, l]. This characterizes thesequences of distinct, positive real numbers for which span{l, x^₁, x^₂, ...} is dense in C[0,1]. The novelty of this result is the treatment of the mostdifficult case when inf_ii = 0 while sup_ii = . The paper settlesthe L and L₁ cases of the following. THEOREM (Full Müntz Theorem in L_p[0,1]). Let p [l, ].Suppose that is a sequence of distinct real numbers greater than –1/p. Then span{x^₀,x^₁, ...} is dense in L_p[0, 1] if and only if      

Projective Modules over the Non-Commutative Sphere

The positive cone of the K₀-group of the non-commutative sphereB is explicitly determined by means of the four basic unboundedtrace functionals discovered by Bratteli, Elliott, Evans andKishimoto. The C*-algebra B is the crossed product A x Z₂ ofthe irrational rotation algebra A by the flip automorphism defined on the canonical unitary generators U, V by (U) = U*,(V) = V*, where VU = e²ⁱ UV and is an irrational real number.This result combined with Rieffel's cancellation techniquesis used to show that cancellation holds for all finitely generatedprojective modules over B. Subsequently, these modules are determinedup to isomorphism as finite direct sums of basic modules. Italso follows that two projections p and q in a matrix algebraover B are unitarily equivalent if, and only if, their vectortraces are equal: [p] = [q]. These results will have the following ramifications. They areused (elsewhere) to show that the flip automorphism on A isan inductive limit automorphism with respect to the basic buildingblock construction of Elliott and Evans for the irrational rotationalgebra. This will, in turn, yield a two-tower proof of thefact that B is approximately finite dimensional, first provedby Bratteli and Kishimoto.     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.