Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

Properties of certain algebras between L∞and H∞

The main result of this paper is that if F is a closed subset of the unit circle, then $\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}\text{H}{}^{\mathrm{\infty }}$ is an M-ideal of $\text{L}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}$. Consequently, if $\text{? ∈ L}{}^{\mathrm{\infty }}$ then ? has a closest element in H^∞ + L_F^∞. Furthermore, if $\text{¦F¦ >0}\text{then}\text{L}{}^{\mathrm{\infty }}\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}$ is not the dual of any Banach space.     

The Poisson transform and representations of a free group

Let G be a free group with r generators, 1 < r < ∞. All the eigenfunctions of an operator on G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations π_z of G on functions on the boundary Ω is considered, defined by $\text{π}{}_{\mathrm{z}}\text{(x)?(ω) = p}{}^{\mathrm{z}}\text{(x, ω)?(x}{}^{\mathrm{?1}}\text{ω)}$, where p(x, ω) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 < s = Re z < 1, π_z is uniformly bounded on an appropriate Hilbert space $\text{H}{}_{\mathrm{s}}\text{(Ω)}$. Finally the uniform boundedness of other special representations of G, obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL(2, Q_p) is proved.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sufficiency and invariance

Suppose that a statistical decision problem is invariant under a group of transformations g?G. T (X) is equivariant if there exists $\text{g}{}^1\text{? G}{}^1$ such that $\text{T(g(X)) = g}{}^1\text{(T((X))}$. We show that the minimal sufficient statistic is equivalent and that if T(X) is an equivariant sufficient statistics and d(X) is invariant under G, then $\text{d}{}^1\text{(T) = Ed(X)∥T}$ is invariant under $\text{G}{}^1$.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.     

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with $\text{E}{}^1{}_1\text{? H}{}^1\text{(X;}\text{Z}{}_2\text{)}$ and $\text{E}{}^1\text{∞ = 0}$. As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy $\text{R}\text{P}{}^{\mathrm{n+k}}\text{R}\text{P}{}^{\mathrm{n}}$, then X will not admit a fixed point free involution.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Spectral representations for abelian automorphism groups of von Neumann algebras

Let $\text{A}$ be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as ¹-automorphisms of $\text{A}$. Let M(σ) be the Banach algebra of bounded linear operators on $\text{A}$ generated by ∝ σ_tdμ(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) $\text{A}$ has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σ_t is inner. It is shown that the Banach algebra L(σ) generated by $\text{∝ ?(t)σ}{}_{\mathrm{t}}\text{dt (? ? L}{}^1\text{(G))}$ is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on $\text{A}$. Also the spectral subspaces of σ are given in terms of projections.     

Profinite groups of finite cohomological dimension

Let 1→N→G→G/N→1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n:=cd_p(G)<∞ and if the order of $\text{H}{}^{\mathrm{k}}\text{(N,}\text{F}{}_{\mathrm{p}}\text{)}$ is finite for k:=cd_p(N), the virtual cohomological p-dimension of G/N equals n?k. To cite this article: T. Weigel, P. Zalesskii, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On a theorem of Osborne and Warner. Multiplicities in the cuspidal spectrum

Osborne and Warner have given a formula for the multiplicity of an integrable discrete series representation in $\text{L}{}^2\text{(}\text{G}\text{Γ}\text{)}$ when G is real rank one semi-simple Lie group and Γ is a discrete subgroup of co-finite volume. We simplify and evaluate this formula to show that for most G (as above) the multiplicity is the formal degree of the representation times the volume of $\text{G}\text{Γ}$. When it is not we give a simple interpretation of the difference.     

Representations of differential operators on a Lie group

In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ as a Lie algebra then we show that the space of C^∞-vectors for U is precisely equal to the C^∞-vectors for the closure $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. If X₀ = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ in the real linear span of the X_j's. An application to quantum field theory is given.Finally, the new characterization of the C^∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C^∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for $\text{G}$ (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators.     

A multiplication of distributions

If Ω denotes an open subset of $\text{R}$ⁿ (n = 1, 2,…), we define an algebra $\text{g}$ (Ω) which contains the space $\text{D}$′(Ω) of all distributions on Ω and such that $\text{C}{}^{\mathrm{\infty }}\text{(Ω)}$ is a subalgebra of $\text{G}$ (Ω). The elements of $\text{G}$ (Ω) may be considered as “generalized functions” on Ω and they admit partial derivatives at any order that generalize exactly the derivation of distributions. The multiplication in $\text{G}$(Ω) gives therefore a natural meaning to any product of distributions, and we explain how these results agree with remarks of Schwartz on difficulties concerning a multiplication of distributions. More generally if q = 1, 2,…, and $\text{?∈O}{}_{\mathrm{M}}\text{(}\text{R}{}^{\mathrm{2q}}\text{)}$—a classical Schwartz notation—for any G₁,…,G_q∈G(σ), we define naturally an element $\text{?G}{}_1\text{,…,G}{}_{\mathrm{q}}\text{∈G(σ)}$. These results are applied to some differential equations and extended to the vector valued case, which allows the multiplication of vector valued distributions of physics.     

Scalar irreducibility of certain eigenspace representations

A sufficient condition for scalar irreducibility of a representation of a group on a topological vector space of the form $\text{{? ∈ C}{}^{\mathrm{\infty }}\text{(R}{}^{\mathrm{\infty }}\text{)∣p}{}_{\mathrm{\infty }}\text{(?)? = 0}\text{for}\text{j=1,…,r}}$ where p₁,…,p_r are polynomials is given. Applications include the differential operators that are invariant under the Cartan motion group of a symmetric space and the Laplace operator.     

Trace formulas for a class of Toeplitz-like operators II

Let P_T denote the orthogonal projection of L²(R¹, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure $\text{dΔ(γ) = ¦ h(γ)¦}{}^2\text{dγ}$, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if $\text{(γ}{}^2\text{+ 1)}{}^{\mathrm{?1}}\text{log}\text{¦ h(γ)¦}$ is summable, if $\text{¦ h ¦}{}^{\mathrm{?2}}$ is locally summable, and if $\text{h}\text{h}{}^{\mathrm{\#}}$ belongs to the span in L^∞ of e^?iyTH^∞:T ? 0, in which h is chosen to be an outer function and h^#(γ) agrees with the complex conjugate of h(γ) on the line, then

$\text{lim trace}\text{T↑∞}\text{{(P}{}_{\mathrm{T}}\text{GP}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? P}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{P}{}_{\mathrm{T}}\text{}}$

exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.     

The unitary group over the integers of a quaternion algebra

Let L be a lattice over the integers of a quaternion algebra with center K which is a $\text{B}$-adic field. Then the unitary group U(L) equals its own commutator subgroup $\text{Ω(L)}$ and is generated by the unitary transvections and quasitransvections contained in it. Let g be a tableau, U(g), U⁺(g), $\text{Ω(g)}$, T(g) be the corresponding congruence subgroups of order g. Then $\text{U(g)}\text{U}{}^{\mathrm{+}}\text{(g)}\text{? X}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{\tau }}\text{Z}{}_2$, and $\text{Ω(g) = T(g)}$ (the subgroup generated by the unitary transvections and quasitransvections with order ≤ g). Let G be a subgroup of U(L) with o(G) = g, then G is normal in U(L) if and only if U(g) ? G ? T(g).     

Unitarization of symplectics and stability for causal differential equations in Hilbert space

Let Sp($\text{H}$) be the symplectic group for a complex Hibert space $\text{H}$. Its Lie algebra sp($\text{H}$) contains an open invariant convex cone C₀; each element of C₀ commutes with a unique sympletic complex structure. The Cayley transform $\text{C}$: X∈ sp($\text{H}$)→(I + X)¹∈ Sp($\text{H}$) is analyzed and compared with the exponential mapping. As an application we consider equations of the form $\text{(}\text{d}\text{dt}\text{) S = A(t)S,}\text{where}\text{t → A(t) ?}\text{C}\text{?}{}_0$ is strongly continuous, and show that if ∝_?∞^∞ ∥A(t)∥ dt < 2 and ∝_{? t8}^∞A(t) dt?C₀, the (scattering) operator

, where S_t′(t) is the solution such that S_t′(t′) = I, is in the range of $\text{B}$ restricted to C₀. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp($\text{H}$) to a unitary operator.     

Primary projections on L2 of a nilmanifold

Let N denote a connected, simply connected nilpotent Lie group with discrete cocompact subgroup Γ. Let U denote the quasi-regular representation on N on $\text{L}{}^2\text{(}\text{N}\text{Γ}\text{)}$. $\text{L}{}^2\text{(}\text{N}\text{Γ}\text{)}$ can be written as a direct sum of primary subspaces with respect to U. A realization for the projections of $\text{L}{}^2\text{(}\text{N}\text{Γ}\text{)}$) onto these primary summands is given in this paper.     

A remark to the preceding paper by Chernoff

It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C₀^∞(R^m) of all positive powers of the Schrödinger operator T = ? Δ + q if q real and in C^∞(R^m) and if $\text{T ? ?a ? b ¦ x ¦}{}^2\text{on}\text{C}{}_0{}^{\mathrm{\infty }}\text{(R}{}^{\mathrm{m}}\text{)}$.     

The number of Hamiltonian circuits

G = 〈V(G), E(G)〉 denotes a directed graph without loops and multiple arrows. $\text{K}$(G) denotes the set of all Hamiltonian circuits of G. Put H(n, r) = max{|E(G)|, |V(G)| = n, 1 ≤ |$\text{K}$(G)| ≤ r}. Theorem: $\text{H(n, 1) = (}\text{n}{}^2\text{2}\text{) + (}\text{n}\text{2}\text{) ?1}$. Further, H(n, 2),…, H(n, 5) are given.