Extensions of unbounded 1-derivations in UHF C1-algebras

We consider unbounded ¹-derivations δ in UHF-C¹-algebras A=(∪^∞_n=1A_n)^?) with dense domain. If ?_n:A→A_n denotes the conditional expectations onto the finite type I factors $\text{A}$_n, then we introduce a weak-commutativity condition for δ and the sequence (?_n). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: $\text{R}$ → Aut($\text{A}$), of ¹-automorphisms, i.e., $\text{δ′(x) =}\text{(d}\text{dt)}\text{α}{}_{\mathrm{t}}\text{(x)¦}{}_{\mathrm{t\; =\; 0}}$ for x?D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ¹-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint.     

Unbounded derivations tangential to compact groups of automorphisms,II

Let G be a compact abelian group, and τ an action of G on a C¹-algebra $\text{U}$, such that $\text{U}$^τ(γ)$\text{U}$^τ(γ)¹ = $\text{U}$^τ(0) $\text{U}$^τ for all $\text{γ ?}\text{G}\text{?}$, where $\text{U}$^τ(γ) is the spectral subspace of $\text{U}$ corresponding to the character γ on G. Derivations δ which are defined on the algebra $\text{U}$_F of G-finite elements are considered. In the special case δ¦_{$\text{U}$^τ} = 0 these derivations are characterized by a cocycle on $\text{G}\text{?}$ with values in the relative commutant of $\text{U}$^τ in the multiplier algebra of $\text{U}$, and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded. Under various restrictions on G and τ properties of the cocycle are deduced which again give characterizations of δ in terms of decompositions into generators of one-parameter subgroups of τ(G) and approximately inner derivations. Finally, a perturbation technique is devised to reduce the case δ($\text{U}$_F) ? $\text{U}$_F to the case δ($\text{U}$_F) ? $\text{U}$_F and δ¦_{$\text{U}$^τ} = 0. This is used to show that any derivation δ with D(δ) = $\text{U}$_F is wellbehaved and, if furthermore G = T¹ and δ($\text{U}$_F) ? $\text{U}$_F the closure of δ generates a one-parameter group of ¹-automorphisms of $\text{U}$. In the case G = T^d, d = 2, 3,… (finite), and δ($\text{U}$_F) ? $\text{U}$_F it is shown that δ extends to a generator of a group of ¹-automorphisms of the σ-weak closure of $\text{U}$ in any G-covariant representation.     

Relative locality of derivations

Let H and K be symmetric linear operators on a C¹-algebra $\text{U}$ with domains D(H) and D(K). H is defined to be strongly K-local if $\text{ω(K(A)}{}^1\text{K(A)) = 0}$ implies $\text{ω(H(A)}{}^1\text{H(A)) = 0}$ for A?D(H) ∩ D(K) and ω in the state space of $\text{U}$, and H is completely strongly K-local if $\text{Ω(K(A)}{}^1\text{K(A))=0}$ implies $\text{Ω(H(A)}{}^1\text{H(A))=0}$ for A ∈ D(H) ∩ D(K) and Ω in the state of $\text{U}$, and H is cpmpletely strongly K-local if $\text{H??}{}_{\mathrm{n}}$ is $\text{K??}{}_{\mathrm{n}}$-local on U?M_n for all n ? 1, where $\text{1}$_n is the identity on the n × n matrices M_n. If $\text{U}$ is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of ¹-automorphisms of $\text{U}$ with generator δ₀ and δ is a derivation which commutes with τ and is completely strongly δ₀-local then δ generates a group α of ¹-automorphisms of $\text{U}$. Various characterizations of α are given and the particular case of periodic τ is discussed.     

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).     

Harmonic analysis on unitary groups

A theory of harmonic analysis on a metric group (G, d) is developed with the model of U$\text{U}$, the unitary group of a C¹-algebra $\text{U}$, in mind. Essential in this development is the set $\text{G}\text{?}{}_{\mathrm{d}}$ of contractive, irreducible representations of G, and its concomitant set P_d(G) of positive-definite functions. It is shown that $\text{G}\text{?}{}_{\mathrm{d}}$ is compact and closed in $\text{G}\text{?}$. The set $\text{G}\text{?}{}_{\mathrm{d}}$ is determined in a number of cases, in particular when G = U($\text{U}$) with $\text{U}$ abelian. If $\text{U}$ is an AW¹-algebra, it is shown that $\text{G}\text{?}$_d is essentially the same as $\text{U}\text{?}$. Unitary groups are characterised in terms of a certain Lie algebra $\text{g}$_u and several characterisations of G = U($\text{U}$) when $\text{U}$ is abelian are given.     

Levi-flat extensions from a part of the boundary

Let G be a bounded domain in $\text{C}\text{×}\text{R}$ such that $\text{G×}\text{R}\text{?}\text{C}{}^2$ is strictly pseudoconvex and U an open subset of bG. We define an open subset $\text{Ω}{}^{\mathrm{U}}$ of $\text{G}$ with the property $\text{Ω}{}^{\mathrm{U}}\text{∩bG=U}$ such that the following extension theorem holds true: for every ?∈C(U) there exist two functions $\text{Φ}{}^{\mathrm{\pm }}\text{∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ^±|_U=? and the graphs Γ(Φ^±) of Φ^± are Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$. Moreover, for each $\text{Φ∈C(Ω}{}^{\mathrm{U}}\text{)}$ such that Φ|_U=? and Γ(Φ) is Levi-flat over $\text{Ω}{}^{\mathrm{U}}\text{∩G}$ one has Φ^??Φ?Φ⁺ on $\text{Ω}{}^{\mathrm{U}}$. We also show that if G is diffeomorphic to a 3-ball and U is the union of simply-connected domains each of which is contained either in the “upper” or in the “lower” part of bG (with respect to the u-direction), then $\text{Ω}{}^{\mathrm{U}}$ is the maximal domain of Levi-flat extensions for some function ?∈C(U). To cite this article: N. Shcherbina, G. Tomassini, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Recurrence relations based on minimization and maximization

Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and g(n) = δ_nI. (2) α + β > 1, min(α, β) > 1, $\text{log}{}_2\text{α}\text{log}{}_2\text{β}$ is rational, and (a) g(n) = δ_n1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases $\text{M(n) = Ω(n}{}^{\mathrm{<mtext>1\; +\; </mtext><mtext>1</mtext><mtext>\gamma </mtext>}}\text{)}$, where γ is defined by α^?γ + β^?γ = 1, and that $\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{M(n)}\text{n}{}^{\mathrm{1\; +\; \gamma }}$ does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max_{1 ? t ? n}(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even.     

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 ? $\text{R}$     

Trace formulas for blocks of Toeplitz-like operators

In this paper we study trace formulas for a class of operators of the form Π_T-GΠ_T in which G designates multiplication by a suitable restricted d × d matrix valued function G(γ) of γ?R¹and Π_T stands for the diagonal d × d matrix (δ_ijP_T) of orthogonal projections P_T of L²(R¹, dδ) onto the space I^T(dδ) of entire functions of exponential type ?T which are square summable on the line relative to the measure $\text{dδ(γ) = ¦h(γ)¦}{}^2\text{dγ}$. It is shown that, for a reasonably large class of h,

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

exists and is independent of the choice of h within the permitted class. These results are then used to study the asymptotic behavior, as T ↑ ∞, of the determinant of I ? Π_TGΠ_T.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

On separating systems of graphs

Given a finite loopless graph G (resp. digraph D), let σ(G), ?(G) and ψ(D) denote the minimal cardinalities of a completely separating system of G, a separating system of G and a separating system of D, respectively. The main results of this paper are:

$\text{δ(G) =}\text{min}\text{m}{}^{\mathrm{m}}{}_{\mathrm{?m/2?}}\text{?γ(G)}\text{and}\text{?(G) = ?}\text{log}{}_2\text{γ(G)?}\text{where}\text{γ(G)}$

denotes the chromatic number of G. (ii) All the problems of determining σ(G), ?(G) and ψ(D) are NP-complete.     

Connected matroids with the smallest whitney numbers

In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if _X(G) = Σμ(0, x)λ^{r ? r(x)} = Σ (?1)^{r ? k}W_kλ^k is the characteristic polynomial of G, then

$\text{w}{}_{\mathrm{k}}\text{?}\text{r}\text{k}\text{+n}\text{r?1}\text{k}$

Thus γ(G) ? 2^{r ? 1} (n+2), where γ(G) = Σw_k. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then

$\text{w}{}_{\mathrm{i}}\text{?}\text{r}\text{i}\text{+ n}\text{r}\text{i+1}\text{for i>1; w}{}_1\text{?r+n}\text{r}\text{2}\text{? 1;}$

|μ| ? (r? 1)n; and γ ? (2^{r ? 1} ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, $\text{r(G) = r(}\text{G}\text{?}\text{) = 4}$, and $\text{n(G) = n(}\text{G}\text{?}\text{) = 3}$ then $\text{(w}{}_1\text{(G))}{}^4{}_{\mathrm{t-G}}\text{? (w}{}_1\text{(}\text{G}\text{?}\text{)) = (8, 20, 18, 7, 1)}$. Further, if β is the Crapo invariant,

$\text{β(G)=}\text{d}{}_{\mathrm{X}}\text{(G)}\text{dλ}\text{(1)}\text{,}$

then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.     

Unbounded derivations commuting with compact group actions. II

Let ($\text{A}$, G, α) be a $\text{C}{}^1$ dynamical system and let δ be a closed ¹ derivation in $\text{A}$ which commutes with α and satisfies $\text{A}$^G ? ker(δ). If $\text{A}$ is a separable Type I $\text{C}{}^1$ algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of ¹ automorphisms of $\text{A}$.     

On the analytic continuation of a certain Dirichlet series

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by

$\text{D}{}_{\mathrm{F}}\text{(s,p,α)}\text{=}\text{∑}\text{α∈Z}{}^{\mathrm{n}}\text{?{0}}\text{F(α)}{}^{\mathrm{?s}}\text{e(ρF(α)+〈α, α〉)}$

where ? ∈ $\text{Q}$, α ∈ $\text{Q}$ⁿ, 〈x, y〉 = x₁y₁ + ? + x_ny_n, e(a) = exp (2πia) for a ∈ $\text{R}$, and s = σ + ti is a complex number. The author proves that: (1) D_F(s, ?, α) has analytic continuation into the whole s-plane, (2) D_F(s, ?, α), ? ≠ 0, is a meromorphic function with at most a simple pole at $\text{s =}\text{n}\text{δ}$. The residue at $\text{s =}\text{n}\text{δ}$ is given explicitly. (3) ? = 0, α ? $\text{Z}$ⁿ, D_F(s, 0, α) is analytic for $\text{α>,}\text{n}\text{(δ ? 1)}$.     

On total matching numbers and total covering numbers of complementary graphs

Best upper and lower bounds, as functions of n, are obtained for the quantities $\text{β}{}_2\text{(G)+β}{}_2\text{(}\text{G}\text{?}\text{)}$ and $\text{α}{}_2\text{(G)+α}{}_2\text{(}\text{G}\text{?}\text{)}$, where β₂(G) denotes the total matching number and α₂(G) the total covering number of any graph G with n vertices and with complementry graph ?.The best upper bound is obtained also for α₂(G)+β₂(G), when G is a connected graph.     

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.     

Non-archimedean group algebras

The group ring R(G) of a group G over a coefficient ring R is well known and so is the L¹ group algebra

$\text{l}{}_1\text{(G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α∈C,∑|α}{}_{\mathrm{g>}}\text{|<∞}$

. We study in this note

$\text{l(R,G)=}\text{∑}\text{g}\text{α}{}_{\mathrm{g}}\text{g:α}{}_{\mathrm{g}}\text{∈R,g∈G,limα}{}_{\mathrm{g}}\text{=0}$

. where R is Z_p(Q_p) the ring (field) of p-adic integers (numbers) equipped with the p-adic valuation. Analogues of certain well known results for group rings and l₁(G) are obtained for l(R,G).