The functor <Emphasis Type="Italic">A</Emphasis><Superscript>min</Superscript> for (<Emphasis Type="Italic">p</Emphasis> − 1)-cell complexes and EHP sequences

Let X be a co-H-space of (p − 1)-cell complex with all cells in even dimensions. Then the loop space ΩX admits a retract Ā ^min(X) that is the evaluation of the functor Ā ^min on X. In this paper, we determine the homology H _*(Ā ^min(X)) and give the EHP sequence for the spaces Ā ^min(X).     

Which compacta are noncommutative ARs?

We give a short answer to the question in the title: dendrits. Precisely we show that the C^*-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C¹ of all (not necessarily commutative) unital C^*-algebras if and only if X is an absolute retract of dimension dimX?1 or, equivalently, that X is a dendrit.     

Homotopy Properties of the Space If(X) of Idempotent Probability Measures

A subspace I_f(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then I_f(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and I_f(X) are equal. It is also proved that, given a compact space X, the compact space I_f(X) is an absolute neighborhood retract if and only if so is X.

     

Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(ε _T ,F _T) where ε _T andF _T are the pull-backs of ε andF toX _T =X x _S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Kom_ε,F) is representable for all ε. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L ^-n ,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F _t) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.     

Hartman-Mycielski functor of non-metrizable compacta

We investigate certain topological properties of the normal functor H, introduced by the first author, which is a certain functorial compactification of the Hartman-Mycielski construction HM. We prove that H is always open and we also find the condition when H X is an absolute retract, homeomorphic to the Tychonov cube.     

Open mapping theorem for spaces of weakly additive homogeneous functionals

We establish that if X and Y are metric compacta and f: X → Y is a continuous surjective mapping, then the openness of the mapping OH(f): OH(X) → OH(Y) of spaces of weakly additive homogeneous functionals is equivalent to the openness of f.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover of X there is a sequence of maps (f _n : X → X) _nεgw such that each f _n is -near to the identity map of X and the family {f _n (X)} _n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z _∞-set in X.      

An end theorem for stratified spaces

We show that a manifold-stratified space X is the interior of a compact manifold-stratified space with boundary if and only if X is tame-ended and a K-theoretic obstruction γ_*(X) vanishes. The obstruction γ_*(X) is a localization of Quinn's mapping cylinder neighborhood obstruction. The main results are Theorem 1.6 and Theorem 1.7 below. In particular, this explains when a G-manifold is the interior of a compact G-manifold with boundary. One of our methods is a new transversality theorem, Theorem 1.16. Oblatum 30-VI-1996 & 21-X-1997 / Published online: 14 January 1999     

On the character and π-weight of homogeneous compacta

UnderGCH, χ(X)≤π(X) for every homogenous compactumX. CH implies that a homogeneous compactum of countable π-weight is first countable. There is a compact space of countable π-weight and uncountable character which is homogeneous underMA+GCH, but not underCH.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Fibrewise construction applied to Lusternik-Schnirelmann category

In this paper a variant of Lusternik-Schnirelmann category is presented which is denoted byQcat(X). It is obtained by applying a base-point free version ofQ=Ω^∞∑^∞ fibrewise to the Ganea fibrations. We provecat(X)≥Qcat(X)≥σcat(X) whereσcat(X) denotes Y. Rudyak’s strict category weight. However,Qcat(X) approximatescat(X) better, because, e.g., in the case of a rational spaceQcat(X)=cat(X) andσcat(X) equals the Toomer invariant. We show thatQcat(X×Y)≤Qcat(X)+Qcat(Y). The invariantQcat is designed to measure the failure of the formulacat(X×S ^r )=cat(X)+1. In fact for 2-cell complexesQcat(X)<cat(X)⇔cat(X×S ^r )=cat(X) for somer≥1. We note that the paper is written in the more general context of a functor λ from the category of spaces to itself satisfying certain conditions; λ=Q, Ω ⁿ Σ ⁿ ,Sp ^∞ orL _f are just particular cases.     

Semistar-Krull and valuative dimension of integral domains

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type ⋆[X] on the polynomial ring D[X], such that, if n := ⋆-dim(D), then n+1 ≤ ⋆[X]-dim(D[X]) ≤ 2n+1. We also establish that if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain, then ⋆[X]-dim(D[X]) = ⋆- dim(D)+1. Moreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dim _v (D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dim _v (D) = n if and only if ⋆[X ₁, . . . , X _n ]-dim(D[X ₁, . . . , X _n ]) = 2n, and that if ⋆-dim _v (D) < ∞ then ⋆[X]-dim _v (D[X]) = ⋆-dim _v (D) + 1. In general ⋆-dim(D) ≤ ⋆-dim _v (D) and equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domains D such that ⋆-dim(D) < ∞ and ⋆-dim(D) = ⋆-dim _v (D). As an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆′)-linked overring T of D, is a ⋆′-Jaffard domain, where ⋆′ is a stable semistar operation of finite type on T. As a consequence of this result we obtain that a Krull domain D, must be a w _D -Jaffard domain.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Classification of Homomorphisms from C (X) to Simple C*-Algebras of Real Rank Zero

Abstract Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K ₀₍ A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism. Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon)     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

On spectral characterizations of amenability

We show that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satisfied: the associated unitary representationπ _X ofG intoL ²(X, μ) is weakly contained into the regular representationλ _G and there exists aG-equivariant norm one projection fromL∞(X×X) ontoL∞(X). We give examples of ergodic discrete group actions which are not amenable, althoughπ _X is weakly contained intoλ _G.     

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600