Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.     

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.     

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.     

Approximately reducing subspaces for unbounded linear operators

We give sufficient conditions for generation of strongly continuous contraction semigroups of linear operators on Hilbert or Banach space. Let L be a dissipative (unbounded) linear operator in a Hilbert space $\text{H}$ and let {P_n} be an increasing sequence of self-adjoint projections converging weakly to the identity projection. We show that if there is a positive integer k such that for all n the range of P_n is contained in the domain of L and mapped by L into the range of P_{n + k}, and if the sequence {LP_n ? P_nLP_n} is dominated in norm (∥LP_n ? P_nLP_n∥ ? a_n) by some {a_n} ? $\text{R}$₊ with ∑_{n = 1}^∞a_n^?1 = ∞, then the closure of the restriction of L to ∪_{n = 1}^∞ range (P_n) is the infinitesimal generator of a strongly continuous contraction semigroup on $\text{H}$. Applications to an important class of finite perturbations, properly larger than the finite Kato perturbations, are given.We also give sufficient conditions for generation of contraction semigroups when {P_γ} (indexed by a directed set) is a set of bounded self-adjoint operators converging weakly to the identity and each having range contained in D(L). In the latter theorem, and in an analogous theorem for dissipative linear operators L in a Banach space, we do not assume that L interchanges at most finitely many of the approximately reducing operators P_γ.     

Unbounded derivations tangential to compact groups of automorphisms

We consider unbounded derivations in C¹-algebras commuting with compact groups of ¹-automorphisms. A closed ¹-derivation δ in a C¹-algebra $\text{U}$ is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈$\text{R}$→τ(t)? Aut($\text{U}$) such that $\text{δ =}\text{d}\text{dt}\text{τ(t)¦}{}_{\mathrm{t\; =\; 0}}$. If δ is known to commute with a compact abelian action α:G→Aut($\text{U}$), and if δ(a) = 0 for all a in the fixed point algebra $\text{U}$^α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that $\text{v(γ)}{}^1\text{= ?v(γ), v(γ)}$ is possibly unbounded but affiliated with the center of {$\text{U}$^α}″, and e^tδ(x) = xe^tv(γ) for all x in the Arveson spectral subspace $\text{U}$^α(γ). In particular, if $\text{U}$ is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group.     

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.     

Toeplitz matrices and determinants with Fisher-Hartwig symbols

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.     

The continuity of derivations of Banach algebras

This paper investigates conditions on a semisimple Banach algebra $\text{U}$ and a Banach $\text{U}$-module $\text{M}$ which insure that every derivation from $\text{U}$ into $\text{M}$ is necessarily a bounded linear operator.     

Abstract kinetic equations relevant to supercritical media

The abstract Hilbert space equation $\text{(T?)′(x) = ?(A?)(x)}$, x∈$\text{R}$₊, is studied with a partial range boundary condition $\text{(Q}{}_{\mathrm{+}}\text{?)(0) = ?}{}_{\mathrm{+}}\text{?}\text{Ran}\text{Q}{}_{\mathrm{+}}$. Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q₊ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.     

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).     

Spectral properties of zeroth-order pseudodifferential operators

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, {E_S}_{S∈$\text{R}$} as a bounded operator on L²(X, dx).Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373–392) and Kato (“Perturbation Theory for Linear Operators,” Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Spec_ESSQ = range(q(x, ξ)) where q is the principal symbol of Q. Turning the attention to the spectral family {E_S}_{S∈$\text{R}$}, it is shown that if $\text{dE}\text{ds}$ is considered as a distribution on $\text{R}$×X×X it is in fact a Lagrangian distribution near the set $\text{{σ=0}?}\text{T}{}^1\text{(}\text{R}\text{×X×X)}\text{0}$ where (s, x, y, σ, ξ,η) are coordinates on T¹($\text{R}$×X×X) induced by the coordinates (s, x, y) on $\text{R}$×X×X. This leads to an easy proof that$\text{?(Q)}$ is a pseudodifferential operator if ?∈C^∞($\text{R}$) and to some results on the microlocal character of E_s. Finally, a look at the wavefront set of $\text{dE}\text{ds}$ leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, ξ).     

Über asymmetrische diophantische approximationen

For irrational numbers θ define α(θ) = lim sup{1/(q(p ? qθ))|p ∈ $\text{Z}$, q ∈ $\text{N}$, p ? qθ > 0} and α(θ) = 0 for rationals. Put $\text{α}\text{(θ) = max{α(θ), α(?0)}}$. Then $\text{U}$ = α($\text{R}$β$\text{Q}$) is an asymmetric analogue to the Lagrange spectrum $\text{U}\text{=}\text{α}\text{(RβQ)}$. Our results concerning $\text{U}$ partly contrast the known properties of $\text{U}$. In fact, $\text{U}$ is a perfect set, each element of which is a condensation point of the spectrum and has continuously many preimages. $\text{U}$ is the closure of its rational elements and of its elements of the form p√m (p ∈ $\text{Q}$), as well. The arbitrarily well approximable numbers form a G_δ-set of 2. category. One has, roughly speaking, $\text{α}\text{→ ∞}$ for α → 1. Finally, the well-known Markov sequence which constitutes the lower Lagrange and Markov spectrum is proved to be a (small) subset of $\text{U}$?[√5,3).     

A variation of Menger's theorem for long paths

In this paper we prove a Mengerian theorem for long paths, namely, that if in order to cut every uv-path of length at least n (n ≥ 2), in a diagraph D, we need to remove at least h points, then there exist {$\text{h}\text{(3n ? 5)}$} interior disjoint uv-paths in D of length at least n. Some variations and applications of this theorem are given as well.     

Classification of certain pairs of operators (P, Q) satisfying [P, Q] = −i Id

In a separable Hilbert space a certain class of pairs of operators (P, Q) satisfying the Born-Heisenberg commutation relation [P, Q] = ?i Id on a dense domain Ω is investigated. This class is essentially defined by requiring Q bounded and self-adjoint, P symmetric and $\text{PΩ ? Ω, QΩ ? Ω}$. We show that Q is absolutely continuous and that P can be thought of as a first order differential operator. The class considered contains the pair “angle ?” and “angular momentum L_z.” It is expected that the methods of this paper can be applied to more general classes of operators (P, Q) including the Schrödinger case.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

A type-B associahedron

The (type-A) associahedron is a polytope related to polygon dissections which arises in several mathematical subjects. We propose a B-analogue of the associahedron. Our original motivation was to extend the analogies between type-A and type-B noncrossing partitions, by exhibiting a simplicial polytope whose h-vector is given by the rank-sizes of the type-B noncrossing partition lattice, just as the h-vector of the (simplicial type-A) associahedron is given by the Narayana numbers. The desired polytope Q^B_n is constructed via stellar subdivisions of a simplex, similarly to Lee's construction of the associahedron. As in the case of the (type-A) associahedron, the faces of Q^B_n can be described in terms of dissections of a convex polygon, and the f-vector can be computed from lattice path enumeration. Properties of the simple dual $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ are also discussed and the construction of a space tessellated by $\text{Q}{}^{\mathrm{<mtext>B</mtext><mtext>1</mtext>}}{}_{\mathrm{n}}$ is given. Additional analogies and relations with type A and further questions are also discussed.     

A note on a theorem of heath

We propose a generalization of Heath's theorem that semi-metric spaces with point-countable bases are developable: A semi-metrizable space X is developabale if (and only if) there is on it a σ-discrete family $\text{C}\text{=?}{}_{\mathrm{m?N}}\text{C}{}_{\mathrm{m}}$ of closed sets, interior-preserving over each member C of which is a countable family {$\text{D}$_n(C): n ∈ N} of collections of open sets such that if U is a neighbourhood of ξ∈X, then there are such a Γ∈$\text{C}$ and such a v∈ N that ξ ? Γ and ξ∈ int ∩ (D: ξ: D∈$\text{D}$_v(Γ))?U.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

Recursive linear orders with recursive successivities

A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities $\text{U}$ is recursively categorical if every recursive linear order with recursive successivities isomorphic to $\text{U}$ is in fact recursively isomorphic to $\text{U}$. We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k₁+g₁+k₂+g₂+…+g_n-1+k_n where each k_n is a finite order type, non-empty for i?{2,…,n-1} and each g_i is an order type from among {ω,ω^*,ω+ω^*}∪{k·η:k<ω}.