Presentations for groups acting on simply-connected complexes

New expansions for global semigroup theory are developed. Many expansions have a left and a right version, each with specific (dual) properties; e.g., the Rhodes expansions ?^$\text{L}$, resp. ?^$\text{R}$, have unambiguous $\text{L}$-resp. $\text{R}$-order. In applications one sometimes needs expansions having both properties simultaneously; these can be constructed by alternately applying the left and the right expansion (possibly infinitely often) while keeping the same set of generators. Thus one obtains an expansion which is invariant under application of the old two expansions and thus has the properties of both (e.g., one obtains -+ with

, and so -+ has unambiguous $\text{L}$-and $\text{R}$-order). It is proved that, in the case of the Rhodes expansion, the new expansion is ‘close’ to the original semigroup; in particular (and this is the main result of the paper), ?⁺_A is finite (resp. finite $\text{J}$-above) if S is finite (resp. finite$\text{J}$-above).     

Non-cancellation and a related phenomenon for the lens spaces

THIS PAPER investigates the structure of the semigroup Σ generated by a set S of non-cancellation examples in the homotopy category. The featured spaces are the 3-dimensional Lens spaces L_p.q. Their products L_p.q × S³ with the 3-sphere S³ are shown to have the same simple-homotopy type, while their own products are shown to determine a unique-division semigroup.     

On the theory of semigroups of operators on locally convex spaces

The behavior of strongly continuous one-parameter semigroups of operators on locally convex spaces is considered. The emphasis is placed on semigroups that grow too rapidly to be treated by classical Laplace transform methods.A space

of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent $\text{R}$ of an operator A on E is defined as an operator on

. It is noted that $\text{R}$ may exist when the classical resolvent (λ ? A)^?1 fails to exist. Conditions on $\text{R}$ are given that are necessary and sufficient to guarantee that A is the generator of a semigroup T(t). The action of $\text{R}$ is characterized by convolution against the semigroup, and the semigroup is computed as the limit of $\text{R}$ acting on an approximate identity.Conditions on an operator B are introduced that are sufficient to guarantee that A + B is the generator of a semigroup whenever A is. A formula is given for the perturbed semigroup.Two characterizations of semigroups that can be extended holomorphically into some sector of the complex plane are given. One is in terms of the growth of the derivative $\text{(}\text{d}\text{dt}\text{) T(t)}$ as t approaches 0, the other is in terms of the behavior of $\text{R}$ⁿ, the powers of the generalized resolvent.Throughout, the generalized resolvent plays a role analogous to the role of the classical resolvent in the work of Hille, Phillips, Yosida, Miyadera, and others.     

Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp

The equation $\text{Lu = ?;(x, u)}\text{on}\text{B × (0, ∞), B}$ bounded, smooth domain in $\text{R}$ⁿ with nonlinear boundary conditions $\text{?u}\text{?v}\text{= g(x, u)}\text{on}\text{?B × (0, ∞)}$ is studied, L being the uniformly parabolic operator with time independent coefficients. Under suitable conditions on the nonlinearities (that do not involve monotonicity) global existence, uniqueness, compactness of the orbits and certain regularizing effects of the semigroup are established. In the case that L is in divergence form it is shown that under generic conditions orbits tend, as t → + ∞, to some equilibrium and that the stable equilibria attract essentially (Baire category) the whole space L²(B).     

A fixed point theorem for eventually nonexpansive semigroups of mappings

Let K be a subset of a Banach space X. A semigroup $\text{F}$ = {?_α ∥ α ∞ A} of Lipschitz mappings of K into itself is called eventually nonexpansive if the family of corresponding Lipschitz constants $\text{{k}{}_{\mathrm{\alpha }}\text{¦ α ? A}}$ satisfies the following condition: for every ? > 0, there is a γ?A such that k_β < 1 + ? whenever ?_β ? ?_γ$\text{F}$ = {?_gg?_α ¦ ?_α ? $\text{F}$}. It is shown that if K is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $\text{F}$:K → K is an eventually nonexpansive, commutative, linearly ordered semigroup of mappings, then $\text{F}$ has a common fixed point. This result generalizes a fixed point theorem by Goebel and Kirk.     

Subnormal weighted translation semigroups

A weighted translation semigroup {S_t} on L²($\text{R}$₊) is defined by $\text{(S}{}_{\mathrm{t}}\text{f)(x) =}\text{(φ(x)}\text{φ(x ? t)}\text{)f(x ? t)}$ for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on $\text{R}$₊. It is shown that {S_t} is subnormal if and only if φ² is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed.     

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.     

Quasimartingales on partially ordered sets

The concept of a quasimartingale, and therefore also of a function of bounded variation, is extended to processes with a regular partially ordered index set V and with values in a Banach space. We show that quasimartingales can be described by their associated measures, defined on an inverse limit space $\text{S × Ω}$ containing $\text{V × Ω}$, furnished with the σ-algebra $\text{P}$ of the predictable sets. With the help of this measure, a Rao-Krickeberg and a Riesz decomposition is obtained, as well as a convergence theorem for quasimartingales. For a regular quasimartingale X it is proven that the spaces ($\text{S × Ω}$, $\text{P}$) and the measures associated with X are unique up to isomorphisms. In the case V = $\text{R}$₊ⁿ we prove a duality between classical (right-) quasimartingales and left-quasimartingales.     

About DK-like spaces and some applications

Let $\text{CSK}$ be the class of all $\text{K}$-scattered spaces having countable ranks. It is shown in this paper that if X is a regular θ-refinable space, then player one has a winning strategy in $\text{G(}\text{DK}\text{,X)}$ if and only if he has one in $\text{G(}\text{CSK}\text{,X)}$. This partly answers Y. Yajima's problem: By topological games, I prove that hereditary disconnectedness, zero-dimensionality and strong zero-dimensionality are equivalent in the realm of non-empty normal compact-scattered weak $\text{θ}$-refinable spaces. A collectionwise normal ultraparacompact-like space is an ultraparacompact space.     

The free fermion field as a Markov field

A definition of a Markov field is given which allows for noncommuting fields. In the commutative case, we recover Nelson's definition (E. Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis12 (1973), 97–112). Conditional expectations are shown to exist in a regular probability gage space, and, using an independence property of these in the free fermion gage space, it is shown that the free fermion field over $\text{H}$^?1($\text{R}$^d) is a Markov field.     

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.     

Fuzzy addition in the L-fuzzy real line

Under the hypothesis L is a chain, we construct a binary operation ⊕ on the L-fuzzy real line $\text{R}$(L) which reduces to the usual addition on $\text{R}$ if ⊕ is restricted to the embedded image of $\text{R}$ in $\text{R}$(L), which yields a partially ordered, abelian cancellation semigroup with identity, and which is jointly fuzzy continuous on $\text{R}$(L). We show ⊕ is unique, i.e. it is the only extension of addition to $\text{R}$(L) which is consistent. We study the relationship between ⊕ and other fuzzy continuous extensions of the usual addition. We also show that fuzzy translation is a weak fuzzy homeomorphism and, under certain conditions, a fuzzy homeomorphism.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

On the generators of semisimple lie algebras

It will be shown that given any element X in a simple Lie algebra $\text{Q}$ over C, there exists a Y ∈ $\text{Q}$ such that the Lie algebra generated by X and Y is $\text{Q}$. The result is extended to the real semisimple Lie algebras. In some sense the main theorem of this paper can be regarded as an extension of Morozov-Jacobson theorem concerning three dimensional simple Lie algebras (see the remark at the end of Sec. 4). A new property of a special class of regular elements, known as the cyclic elements, is given.     

On the left regular representation of a separable locally compact group

For an arbitrary separable locally compact group G we exhibit a canonical Borel subset $\text{G}\text{?}{}_{\mathrm{\Delta }}$ of the quasi-dual $\text{G}\text{?}\text{of}\text{G}$ (with the Mackey Borel structure), such that $\text{G}\text{?}{}_{\mathrm{\Delta }}$ is a standard Borel space in the induced Borel structure, and such that the canonical measure for the left regular representation λ_Gof G is concentrated on $\text{G}\text{?}{}_{\mathrm{\Delta }}$. On the basis of this we discuss the (non-unimodular) “Plancherel theorem.”     

Compact abelian groups of automorphisms of von Neumann algebras

Let G be a compact abelian group, acting σ-weakly continuously as a group of 1-automorphisms α on a von Neumann algebra $\text{R}$. We give necessary and sufficient conditions for α to be inner, based on the structure of the lattice of projections in the center of the fixed-point algebra. As an application, we show that if α satisfies a spectrum condition with respect to a suitably chosen positive semigroup in the dual of G, then α is inner, and the implementing unitary representation can be chosen with positive spectrum.     

Unitary block designs

Given a projective plane $\text{E}$ over the field of q² elements and a unitary polarity π of $\text{E}$ it is possible to construct the well-known unitary design $\text{U}$ whose points are the absolute points of π and whose blocks are the non-absolute lines of π. A relation of perpendicularity is defined between blocks and it is shown that this relation can be described in terms of the incidence structure of $\text{U}$. The projective plane $\text{E}$ together with the polarity π can then be reconstructed from the design $\text{U}$ in such a way that any automorphism of $\text{U}$ extends to a collineation of $\text{U}$ which commutes with π.     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Reflexive algebras with finite width lattices: Tensor products,cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ¹-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg($\text{L}$₁ ? $\text{L}$₂) = Alg $\text{L}$₁ ? Alg $\text{L}$₂, where each $\text{L}$_i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each $\text{L}$_i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if $\text{L}$ is a finite-width commutative subspace lattice and $\text{K}$ is the set of compact operators then the quasitriangular algebra Alg $\text{L}$ + $\text{K}$ is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.