Extensions of factorial states of C1-algebras

Let A be a $\text{C}{}^1$-algebra, B be a $\text{C}{}^1$-subalgebra of A, and φ be a factorial state of B. Sometimes, φ may be extended to a factorial state of A by a tensor product method of Sakai (“$\text{C}{}^1$-algebras and $\text{W}{}^1$-algebras, Springer-Verlag, Berlin/Heidelberg/ New York 1971”). Sometimes, there is a weak expectation of A into $\text{π}{}_{\mathrm{\phi }}\text{(B)}$, and then factorial extensions may be found by a method of Sakai and Tsui (Yokohama Math. J.29 (1981), 157–160). These two methods are shown to have the same effect, and the factorial extensions produced by them are analysed.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

A Dunford-Pettis theorem for L1H∞⊥

The spaces in the title are associated to a fixed representing measure m for a fixed character on a uniform algebra. It is proved that the set of representing measures for that character which are absolutely continuous with respect to m is weakly relatively compact if and only if each m-negligible closed set in the maximal ideal space of L^∞ is contained in an m-negligible peak set for H^∞. J. Chaumat's characterization of weakly relatively compact subsets in $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ therefore remains true, and $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ is complete, under the first conditions. In this paper we also give a direct proof. From this we obtain that $\text{L}{}^1\text{H}{}^{\mathrm{\infty \perp }}$ has the Dunford-Pettis property.     

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n₁((q¹ ? 1)/N), where N|q ? 1, gcd(n₁,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n₁l, k = (n₁ ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n₁, m) linear code, $\text{V}{}^1{}_{\mathrm{d}}$ over GF(q^d) where d = gcd (ord_n1(q), l). Specifically we study A(z) when $\text{V}{}^1{}_{\mathrm{d}}$ is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A_μ which completely determine the weight distributionof any irreducible cyclic (n₁(2¹ ? 1), k) code over GF(2) for all n₁ ? 17.     

The covering problem for the fields Q(−1)12 and Q(−3)12

Let k be a positive square free integer, $\text{N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ the ring of algebraic integers in $\text{Q(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and S the unit sphere in C_n, complex n-space. If A₁,…, A_n are n linearly independent points of C_n then L = {u₁Au + … + u_nA_n} with $\text{u}{}_{\mathrm{r}}\text{∈ N(?k)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is called a k-lattice. The determinant of L is denoted by d(L). If L is a covering lattice for S, then $\text{θ(S, L) =}\text{V(S)}\text{d(L)}$ is the covering density. L is called locally (absolutely) extreme if θ(S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.     

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.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.     

Loeb-measurable solutions to 1finite games

Let ¹M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal ¹finite subset of ¹N such that $\text{F = {1,…,γ}, γ?}{}^1\text{N?N}$, we define a class of ¹finite cooperative games having the form $\text{Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν) = 〈F,A(F),}{}^1\text{ν〉}$, where A(F) is the internal algebra of the internal subsets of F, and ${}^1\text{ν}$ is a set-function with $\text{Dom}{}^1\text{ν=A(F)}$, $\text{Rng}{}^1\text{ν =}{}^1\text{R}{}_{\mathrm{+}}$, and ${}^1\text{ν(Ø) = 0}$. If $\text{SI(}{}^1\text{ν)}$ is the space of S-imputations of a game Γ_F(¹ν) such that ${}^1\text{ν(F)<η}$, for some $\text{η?}{}^1\text{N}$, then we prove that $\text{SI(}{}^1\text{ν)}$ contains two nonempty subsets: $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$, termed the quasi-kernel and S-bargaining set, respectively. Both $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ are external solution concepts for games of the form Γ_F (¹ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L(Θ) is the Loeb space generated by the ¹finitely additive measure space 〈F, A(F), U_F〉, and if a game Γ_F(¹ν) has a nonatomic representation $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ on L(Θ) with respect to S-bounded transformations, then the standard part of any element in $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ is Loeb-measurable and belongs to the quasi-kernel of $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ defined in standard terms.     

Derivations on a C1-algebra and its double dual

Let A be a C¹-algebra and X a Banach A-module. The module action of A on X gives rise to module actions of $\text{A}{}^7$ on $\text{X}{}^1$ and $\text{X}{}^7$, and derivations of A into X (resp. $\text{X}{}^1$) extend to derivations of $\text{A}{}^7$ into $\text{X}{}^7$ (resp. $\text{X}{}^1$). If A is nuclear, and X is a dual Banach A-module with $\text{X}{}^1$ weakly sequentially complete, then every derivation of A into X is inner. Under the same hypothesis on A, the extension to the finite part of $\text{A}{}^7$ of any derivation of A into any dual Banach A-module is inner, as are all derivations of A into $\text{A}{}^1$. Every derivation of a semifinite von Neumann algebra into its predual is inner.     

A Hamiltonian decomposition of K2m1, 2m ≥ 8

It is shown that $\text{K}{}_{\mathrm{2m}}{}^1$, 2m ≥ 8, can be decomposed into Hamiltonian circuits. A direct construction utilizing difference methods is given for 2m ≡ 0 (mod 4). The case 2m ≡ 2 (mod 4) is handled inductively by means of a construction which shows that $\text{K}{}_{\mathrm{4m\; ?\; 2}}{}^1$ admits such a decomposition if $\text{K}{}_{\mathrm{2m}}{}^1$ does.     

On the expansion of Cϱ1(V + I) as a sum of zonal polynomials

The coefficients a_τ?, sometimes called “generalized binomial coefficients” in the expansion $\text{C}{}_{\mathrm{?}}{}^1\text{(V +I) = Σ}{}_{\mathrm{\tau }}\text{a}{}_{\mathrm{?\tau }}\text{C}{}_{\mathrm{\tau }}{}^1\text{(V)}$, are computed explicitly when t = r + 1, where ? is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general a_τ?. Several properties of a_τ? are proved. A connection between the a_τ? and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials.     

The isometries of LP(Ω, X)

Let (Ω, ∑, μ) be a finite measure space and X a separable Banach space. We characterize the linear isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ onto itself for 1 ? p < ∞, p ≠ 2 under the condition that X is not the l^p-direct sum of two nonzero spaces (for the same p). It is shown that T is such an isometry if and only if (Tf)(·) = S(·)h(·)(Φ(f))(·), where Φ is a set isomorphism of ∑ onto itself, S is a strongly measurable operator-valued map such that S(t) is a.e. an isometry of X onto itself, and h is a scalar function which is related to Φ. It is further shown that for a big class of measure spaces (perhaps all nontrivial ones) the condition on X is also a necessary condition for the above conclusion to hold. In the case when X is a Hilbert space the injective isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ are also characterized. They have the same form as above, except that Φ and S(t) are not necessarily onto.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Transformation group C1-algebras with continuous trace

We obtain several results characterizing when transformation group C¹-algebras have continuous trace. These results can be stated most succinctly when ($\text{G, Ω}$) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if the stability groups vary continuously on Ω and compact subsets of Ω are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if ($\text{G, Ω}$) is not second countable, that the natural maps of $\text{G}\text{S}{}_{\mathrm{x}}$ onto G · x are homeomorphisms for each x. Then $\text{C}{}^1\text{(G, Ω)}$ has continuous trace if and only if compact subsets of Ω are wandering and an additional C¹-algebra, constructed from the stability groups and Ω, has continuous trace.     

Exposed points and extremal problems in H1

If φ ∈ L_∞, we denote by T_φ the functional defined on the Hardy space H¹ by

$\text{T}{}_{\mathrm{\phi }}\text{(?) =}\text{∫}\text{?π}\text{π}\text{?(e}{}^{\mathrm{i\theta }}\text{) φ(e}{}^{\mathrm{i\theta }}\text{)}\text{dθ}\text{2π}$

. Let S_φ be the set of functions in H¹ which satisfy $\text{T}{}_{\mathrm{\phi }}\text{(?) = ∥T}{}_{\mathrm{\phi }}\text{∥}\text{and}\text{∥? ∥}{}_1\text{? 1}$. It is known that if φ is continuous, then S_φ is weak-¹ compact and not empty. For many noncontinuous φ each S_φ is weak-¹ compact and not empty. A complete descr ption of S_φ if S_φ is weak-¹ compact and not empty is obtained. S_φ is not empty if and only if $\text{S}{}_{\mathrm{\phi }}\text{= S}{}_{\mathrm{\psi }}\text{and}\text{ψ = ¦ ?¦}\text{?}$ for some nonzero ? in H¹. It is shown that if $\text{φ = ¦? ¦}\text{?}$ and $\text{? = pg}$, where p is an analytic polynomial and g is a strong outer function, then S_φ is weak-¹ compact. As the consequence, if $\text{? = p}$, then S_φ is weak-¹ compact.     

Decomposition of Km,n(Km,n1) into cycles (circuits) of length 2k

In this paper we give necessary and sufficient conditions in order that K_m,n ($\text{K}{}_{\mathrm{m,n}}{}^1$) admits a decomposition into 2k-cycles (2k-circuits). This answers conjectures of J. C. Bermond (Thesis, Paris XI (Orsay), 1975) and J. C. Bermond and V. Faber (J. Combinatorial Theory Ser. B21 (1976), 146–155).     

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.     

Powers of matrices and idempotency

In this paper, we establish the following results: Let A be a square matrix of rank r. Then (a) $\text{(A+A}{}^1\text{)}\text{2}$ is idempotent of rank r, and tr_rA (defined as the sum of the principal minors of order r in A) is one iff A is Hermitian idempotent. (b) A^s=A^t for some positive integers s≠t, and trA=rankA iff A is idempotent. (c) $\text{A(A1A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t iff $\text{AA}{}^1\text{=A}{}^1\text{A}$ is idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integers s≠0 iff $\text{AA}{}^1\text{=A}{}^1\text{A}$. (d) $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{=A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{t}}$ for some integers s≠t and rankA=trA iff A is Hermitian idempotent, while $\text{A(A}{}^1\text{A)}{}^{\mathrm{s}}\text{= A}{}^1\text{(AA}{}^1\text{)}{}^{\mathrm{s}}$ for some integer s iff A is Hermitian. Here $\text{A}{}^1$ indicates the conjugate transpose of A, and P^-α is defined iff (P⁺)^α=(P^α)⁺ for all positive integers α and P⁺ is the Moore-Penrose inverse of P.