Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.     

Cesàro asymptotics for the orders of SLk(Zn) and GLk(Zn) as n→∞

Given an integer k>0, our main result states that the sequence of orders of the groups $\text{SL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$ (respectively, of the groups $\text{GL}{}_{\mathrm{k}}\text{(}\text{Z}{}_{\mathrm{n}}\text{)}$) is Cesàro equivalent as n→∞ to the sequence C₁(k)n^k2?1 (respectively, C₂(k)n^k2), where the coefficients C₁(k) and C₂(k) depend only on k; we give explicit formulas for C₁(k) and C₂(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to $\text{n}\text{6}\text{π}{}^2$. We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Packing a convex domain with similar convex domains

Given a convex domain C and a positive integer k, inscribe k nonoverlapping convex domains into C, all of them similar to C. Denote by f(k) the maximal sum of their circumferences. In this paper it is shown, that for C square, parallelogram or triangle (1) the first increase of f(k) after k = l² occurs not later than at k = l² + 2, (2) constructions can be given, where the following lower bounds are attained for f(k) = f(l² + j):

$\text{(}\text{1}\text{c}\text{) ? l +}\text{(j ? 1)}\text{2l}\text{j}\text{odd}\text{, l? 2}$

$\text{? l +}\text{j}\text{2}\text{(l + 1) j}\text{even}\text{, l?2}$

where c denotes the circumference of C.     

Geometric conditions for the existence of positive eigenvalues of matrices

Finite-dimensional theorems of Perron-Frobenius type are proved. For A∈$\text{C}$ⁿⁿ and a nonnegative integer k, we let w_k (A) be the cone generated by A^k, A^k+1,…in $\text{C}$ⁿⁿ. We show that A satisfies the Perron-Schaefer condition if and only if the closure $\text{W}$_k(A) of w_k(A) is a pointed cone. This theorem is closely related to several known results. If k?v₀(A), the index of the eigenvalue 0 in spec A, we prove that A has a positive eigenvalue if and only if w_k(A) is a pointed nonzero cone or, equivalently $\text{W}$_k(A) is not a real subspace of $\text{C}$ⁿⁿ. Our proofs are elementary and based on a method of Birkhoff's. We discuss the relation of this method to Pringsheim's theorem.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

On a duality for crossed products of C1-algebras

Let $\text{U}$ be a C¹-algebra, and G be a locally compact abelian group. Suppose α is a continuous action of G on $\text{U}$. Then there exists a continuous action $\text{}\text{?}$ga of the dual group $\text{G}\text{?}$ of G on the C¹-crossed product by α such that the C¹-crossed product is isomorphic to the tensor product and the C¹-algebra of all compact operators on L²(G).     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

Über die Wurzelschranke für das Minimalgewicht von codes

Let d be the minimum distance of an (n, k) code $\text{C}$, invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that $\text{C}$ contains the repetition code, that dim($\text{C}$ ∏ $\text{C}$^⊥) = k ? 1 and that the supports of the minimal weight vectors of $\text{C}$ form a 2-design. Then d² ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d² ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d² ? d + 1 > n and F = GF(2) then (d ? 1)² ? n. Examples are the generalized quadratic residue codes.     

The visually distinct configurations of k sets

Two sets of sets, C₀ and C₁, are said to be visually equivalent if there is a 1-1 mapping m from C₀ onto C₁ such that for every S, T?C₀, S ∩ T=0 if and only if m(S)∩ m(T)=0 and S?T if and only if m(S)?m(T). We find estimates for V(k), the number of equivalence classes of this relation on sets of k sets, for finite and infinite k. Our main results are that for finite k, $\text{1}\text{2}\text{k}{}^2\text{-k log k <log V (k)<ak}{}^2\text{+βk+log k}$, where α and β are approximately 0.7255 and 2.5323 respectively, and there is a set N of cardinality $\text{1}\text{2}\text{(k}{}^2\text{+k)}$ such that there are V(k) visually distinct sets of k subsets of N.     

Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

On the invariants of some Zi-extensions

Let k ? k₁ ? … ? K be a Z_i-extension. The relations of $\text{λ(}\text{K}\text{k}\text{)}$ and $\text{λ(}\text{KF}\text{F}\text{)}$ is studied, where $\text{F}\text{k}$ is a cyclic l-extension. If $\text{M}\text{k}$ is another Z_i-extension of k, it is shown that for i ? 0$\text{λ(}\text{Mk}{}_{\mathrm{i}}\text{k}{}_{\mathrm{i}}\text{) = rl}{}^{\mathrm{i}}\text{+ C}$, under minimal additional hypotheses. Finally if $\text{MK}\text{k}$ has a unique totally ramified prime, and X_K is cyclic, it is shown that MK can contain at most one Z_i-extension with non-zero μ invariant.     

On some t−(2k, k, λ) designs

t?(2k, k, λ) designs having a property similar to that of Hadamard 3-designs are studied. We consider conditions (i), (ii), or (iii) for t?(2k, k, λ) designs: (i) The complement of each block is a block. (ii) If A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. (iii) if A and B are a complementary pair of blocks, then ∥ A ∩ C ∥ = ∥ B ∩ C ∥ or ∥ A ∩ C ∥ = ∥ B ∩ C ∥ ± u holds for any block C distinct from A and B, where u is a positive integer. We show that a t?(2k, k, λ) design with t ? 2 and with properties (i) and (ii) is a 3?(2u(2u + 1), u(2u + 1), u(2u² + u ? 2)) design, and that a t?(2k, k, λ) design with t ? 4 and with properties (i) and (iii) is the 5-(12, 6, 1) design, the 4-(8, 4, 1) design, a $\text{5?(2u}{}^2\text{, u}{}^2\text{,}\text{1}\text{4}\text{(u}{}^2\text{? 3) (u}{}^2\text{? 4))}$ design, or a $\text{5?(}\text{2}\text{3}\text{u(2u + 1),}\text{1}\text{3}\text{u(2u = 1),}\text{1}\text{5 4}\text{u(2u}{}^2\text{+ u ? 9) (2u}{}^2\text{+ u ? 12))}$ design.     

Quantization of the action of U(k,l) on R2(K + l)

The natural action of U(k, l) on $\text{C}$^{k + l} leaves invariant a real skew non-degenerate bilinear form B, which turns $\text{C}$^{k + l} into a symplectic manifold (M, ω). The polarization F of M defined by the complex structure of $\text{C}$^{k + l} is non-positive. If L is the prequantization complex line bundle carried by (M, ω), then U(k, l) acts on the space $\text{U}$ of square-integrable $\text{L ? ΛF}{}^1$ forms on M, leaving invariant the natural non-degenerate, but non-definite, inner product ((·, ·)) on $\text{U}$. The polarization F also defines a closed, densely defined covariant differential ?? on $\text{U}$ which is U(k, l)-invariant. Let $\text{⊥}$ denote orthocomplementation with respect to ((·, ·)). It is shown that the restriction of ((·, ·)) to the U(k, l)-stable subspace ? (Ker ??) ∩ (Im ??)^$\text{⊥}$ is semi-definite and that the unitary representation of Uk, l on the Hilbert space $\text{H}$ arising from ? by dividing out null vectors is unitarily equivalent to the representation of U(k, l) obtained from the tensor product of the metap ectic and $\text{Det}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ representations of MU(k, l), the double cover of U(k, l).     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

Independence in graphs with maximum degree four

Let $\text{C}$ be the class of triangle-free graphs with maximum degree four. A lower bound for the number of edges in a graph of $\text{C}$ is derived in terms of its order p and independence β. Also a characterization of certain minimum independence graphs in $\text{C}$ is provided. Let r(k) be the smallest integer such that every graph in $\text{C}$ with at least r(k) vertices has independence at least k. The values of r(k) for all k may be derived from our main theorem and $\text{4}\text{13}$ obtained as the best possible lower bound for the independence ratio $\text{β}\text{p}$ of graphs in $\text{C}$.     

C1-algebras without idempotents

The simplest statement of the main results are these: Let π be a free group on 2 generators. Let Cπ be the complex ring and L¹π the ring extension to L¹ sums. Then L¹π contains no idempotents. Furthermore, if α ? Cπ, β?L¹π are nonzero, then αβ ≠ 0. The first result is in the direction of proving that a certain simple $\text{C}{}^1\text{-}\text{algebra}$ has no idempotents yielding a counter-example to the suggestion that simple $\text{C}{}^1\text{-}\text{algebras}$ are generated by their projections.     

Semi-crossed products of C1-algebras

Given a C¹-algebra $\text{U}$ and endomorphim α, there is an associated nonselfadjoint operator algebra $\text{Z}$⁺ X_α$\text{U}$, called the semi-crossed product of $\text{U}$ with α. If α is an automorphim, $\text{Z}$⁺ X_α$\text{U}$ can be identified with a subalgebra of the C¹-crossed product $\text{Z}$⁺ X_α$\text{U}$. If $\text{U}$ is commutative and α is an automorphim satisfying certain conditions, $\text{Z}$⁺ X_α$\text{U}$ is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C₀(S) given by α(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z⁺X_αC₀(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.     

On the ∂-Neumann problem for generalized functions

Using the new theory of generalized functions developed by one of the authors the $\text{?}$ equation in $\text{C}$ⁿ is studied. In particular it is proven that if G is any generalized function on $\text{C}$ (in the above sense) then there is a generalized function S on $\text{C}$ such that $\text{?S}\text{?}\text{z}\text{?}\text{= G}$. Several other results are proven valid in polydiscs of $\text{C}$ⁿ, for which differential forms whose coefficients are generalized functions are introduced.