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We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite $k$-network, respectively. We are motivated by the following result of Corson (1966): If the space $C_{c}(X)$ of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology is a Banach space, then $C_{c}(X)$ endowed with the weak topology is an $\aleph_0$-space if and only if $X$ is countable. We extend Corson's result as follows: If the space $E:=C_{c}(X)$ is a Fr\'echet lcs, then $E$ endowed with its weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $X$ is countable. We obtain a necessary and some sufficient conditions on a Fr\'echet lcs to be an $\aleph$-space in the weak topology. We prove that a reflexive Fr\'echet lcs $E$ in the weak topology $\sigma(E,E')$ is an $\aleph$-space if and only if $(E,\sigma(E,E'))$ is an $\aleph_0$-space if and only if $E$ is separable. We show however that the nonseparable Banach space $\ell_{1}(\mathbb{R})$ with the weak topology is an $\aleph$-space., Comment: 18 pages

We show a new functional limit theorem for weakly dependent regularly varying sequences of random vectors. As it turns out, the convergence takes place in the space of $$\mathbb R ^{d}$$ valued cadlag functions endowed with the so-called weak $$M_{1}$$ topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod’s $$M_1$$ topology is actually necessary for the limit theorem to hold.

We define and study the properties of some subsets of X with respect to a weak structure on X and generalize some already established results.

In this note we investigate quantizations of the weak topology associated with a pair of dual linear spaces. We prove that the weak topology admits only one quantization called the weak quantum topology, and that weakly matrix bounded sets are precisely the min-bounded sets with respect to any polynormed topology compatible with the given duality. The technique of this paper allows us to obtain an operator space proof of the noncommutative bipolar theorem.

The following notations associated with a mapping f : (X, τ) → (Y, σ) have been introduced in [D. S. Janković: A note on mappings of extremely disconnected spaces, Acta Math. Hungar. (1985), 83-92]: where for each x ∊ X. Interrelationships between well-known openness-type properties (a.o.S., w.o., -openness, a.o.W.) for f, f*, fα, fα are studied.

Given a foundation locally compact Hausdorff topological semigroup S , we consider on Ma ( S )* the τc -topology, i.e. the weak topology under all right multipliers induced by measures in Ma ( S ). For such an arbitrary S the τc -topology is not weaker than the weak*-topology and not stronger than the norm topology on Ma ( S )*. However, a further investigation shows that for compact S the norm topology and τc -topology coincide on every norm bounded subset of Ma ( S ). Among the other results we mention that except for discrete S the τc -topology is always different from the norm-topology. Finally, we give some results about τc -almost periodic functionals.

Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra \(\mathfrak M\). We show that there are a surjective ultraweakly continuous ∗-homomorphism \(\Sigma:\mathfrak M\to\mathfrak M\) and a Σ-derivation \(D:\mathfrak M\to\mathfrak M\) such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ-derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ-derivation on \(\mathfrak M\), then there is an ultraweakly continuous linear mapping \(\Sigma:\mathfrak M\to\mathfrak M\) such that d is a ∗-Σ-derivation.

Every selection f : F 2 ( X ) → X for the family F 2 ( X ) of at most two-point subsets of a set X naturally defines an order-like relation ⪯ f on X by letting x ⪯ f y if and only if f ( { x , y } ) = x . In particular, it also defines a natural interval-like topology on X. In the present paper we study properties of this topology, and its possible relations with other topologies on X with respect to which f is Vietoris continuous.

We show that every state ω on a lattice effect algebra E induces a uniform topology on E. If ω is subadditive this topology coincides with pseudometric topology induced by ω. Further, we show relations between the interval and order topology on E and topologies induced by states.

A ternary ring of operators is an off-diagonal corner of a C*-algebra and the predual of a ternary ring of operators (if it exists) is of the form pR * q for some von Neumann algebra R and projections p and q in R. In this paper, we prove that a subspace of the predual of a ternary ring of operators is completely 1-complemented if and only if it is completely isometrically isomorphic to the predual of some ternary ring of operators. We next give an operator space characterization of the preduals of separable injective von Neumann algebras. Finally, we prove some concrete results about the finite dimensional completely 1-complemented subspaces of a von Neumann algebra predual.

It is proved that if a normal semifinite weight ϕ on a von Neumann algebra \(\mathcal{M}\) satisfies the inequality \(\phi (|a_1 + a_2 |) \leqslant \phi (|a_1 |) + \phi (|a_2 |)\) for any selfadjoint operators \(a_1 ,a_2 \) in \(\mathcal{M}\), then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality \(|\phi (a)|{\text{ }} \leqslant {\text{ }}\phi (|a|)\) is refined and reinforced.

Some relations between cardinal invariants ofX andC(X) are established in the weak topology, whereC(X) is the space of continuous real-valued functions onX in the compact-open topology.

In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L ( H ) \mathcal {L}(H) of closed subspaces of a separable Hilbert space H. We compare the order topology on L ( H ) \mathcal {L}(H) with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on L ( H ) \mathcal {L}(H) .

Let f:\((\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)\) be a singularity, that is the germ ofa holomorphic function, with an isolated critical point at the origin. It follows from implicit function theorem that in a neighbourhood of the origin in the space \(\mathbb{C}^{n}\)the level setf-l \({f}^{-1}(\varepsilon)\)for for e≠0 is a non-singular analytic manifold and the level set \({f}^{-1}(0)\) is a nonsingular manifold away from the origin. At the point \( 0 \,\epsilon \,\mathbb{C}^{n}\) the level set has a singular point.

For each real infinitely dimensional Hilbert space X a weakly continuous function f:X → R is constructed, possessing at x 0=0 a strong minimum (i.e. a local minimum with respeel to the norm topology), which is not a weak minimum (i.e. a local minimum with respect to the weak topology). For the given example different convexity properties are discussed.

Introduction This chapter is devoted to hypercyclicity and supercyclicity with respect to the weak topology of a given Banach space. Let X be a separable infinite-dimensional Banach space. An operator T ∈ L( X ) is said to be weakly hypercyclic if it is hypercyclic when considered as an operator on the topological vector space ( X , ω), in other words, if there exists some vector x ∈ ω whose T -orbit O ( x , T ) is weakly dense in X . Such a vector x is of course called a weakly hypercyclic vector for T . One defines in the same way weakly supercyclic operators and weakly supercyclic vectors. One unpleasant fact immediately comes to mind when considering these defi- nitions. Up to now, we have almost exclusively concentrated on hypercyclicity or supercyclicity for linear operators acting on completely metrizable topological vector spaces; however, the weak topology of an infinite-dimensional Banach space is neither Baire nor metrizable. This means first that we have no Hypercyclicity Criterion at our disposal and second that we must be careful with sequences , since a vector z ∈ X may belong to the weak closure of some T -orbit O ( x , T ) without being the weak limit of a sequence ( T nk ( x )). Having said that, the first natural question is whether weak hypercyclicity and supercyclicity really make sense, i.e. whether there exist weakly hypercyclic or supercyclic operators which are not already hypercyclic or supercyclic with respect to the norm topology. Fortunately the answer is positive, but this is a non-trivial result.

The dual of X* is another Banach space X**. This is called the second dual or the bidual of X. Let J be the map from X into X** that associates with x ∈ X the element F x ∈ X** defined as $${F_x}(f) = f(x)\;for\;all\;f \in {X^*}.$$ Then J is a linear map and ‖Jx‖ = ‖x‖. (See (9.2).) Thus J is an isometric imbedding and we can regard X as a subspace of X**.

When we say that a sequence f n in the space C[0, 1] converges to f, we mean that ‖f n − f‖ → 0 as n → ∞; and this is the same as saying f n converges to f uniformly. There are other notions of convergence that are weaker, and still very useful in analysis. This is the motivation for studying different topologies on spaces of functions, and on general Banach spaces.

We give an alternate description of algebras in the class of ultraweakly closed subspaces of q(X) via the preannihilator. We then apply this result to show that proper ultraweakly closed algebras of bounded operators on an infinite-dimensional Hilbert space X have infinite codimension. We also use this alternate description of algebras to say something the structure of rank-one operators in unicellular algebras. We begin with some basic definitions and notation. For ', an infinitedimensional Hilbert space, let 5F(X) denote the set of all bounded linear operators on X and let Y(X) denote the set of all trace-class operators on X'. Then Y('), equipped with the trace norm, is a Banach space whose dual is 5( ). The duality is given by the linear functional on 7(X) x (X) defined by (t, x) tr(tx) for t EzY(X), x E (X), where tr denotes the trace. Thus we get a w*-topology on 5(X), which is also known as the ultraweak topology. The weak-operator topology (which we shall refer to as the weak topology) is actually weaker than the ultraweak topology, so all the results stated in this paper for ultraweakly closed subspaces are true for weakly closed subspaces. An excellent exposition of the role of this duality theory in invariant subspace theory is [ 1]. We shall follow the notation of [ 1], which the reader can consult for more background. As usual, we can use the above duality to define preannihilators. For ? an ultraweakly closed subspace of (X), the preannihilator is X1 = {t EY7(X) I tr(tm) = 0 for all m Ecz}. The codimension of A'(codim(.')) is the vector space dimension of {f(X)/ X'}. If we identify all infinite cardinals, then this is also equal to the vector space dimension of X1 . Given x, y E X, the operator x 0 y is the rank-one operator defined by x X y(z) = (z, y)x for z E X. Received by the editors July 3, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

Let M, N, R be W∗-algebras, with R unitally embedded in both M and N. By using Reduction Theory, we describe the predual (M⊗̄RN)∗ of the W∗-tensor product M⊗̄RN over the common W∗-subalgebra R in the separable predual case. We also analyze the case when R is a direct sum of full matrix algebras, without any separability assumption. In the last situation, the predual (M⊗̄RN)∗ is described by the center Z(RM∗⊗N∗R) of the R–R bimodule RM∗⊗N∗R, the last one being isomorphic to the predual of M⊗̄N. It is also shown that such a reduction-free result cannot be extended to the remaining cases.

It is proved that if a normal semifinite weight ψ on a von Neumann algebra M satisfies the inequality ψ(|a 1 + a 2|) ≤ ψ(|a 1|) + ψ(|a 2|) for any selfadjoint operators a l, a 2 in M, then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality |ψ(a)| ≤ ψ(|a|) is refined and reinforced.

We discuss the domain-theoretic and topological content of the operator calculus used in the Irish School of the Vienna Development Method (VDM♣) of formal systems development. Thus, we examine the Scott continuity, or otherwise, of the basic operators used in this calculus when viewed as operators on the domain (X → Y ) of partial functions mapping X into Y. It turns out that the override, one of the more important of the basic operators, is not Scott continuous, and in order to overcome this problem we introduce another topology, which we call here the strong Cantor topology, by means of the topological tool of convergence classes. Indeed, the strong Cantor topology is the smallest topology which refines the Scott and Lawson topologies and is such that, with respect to it, all the basic operators we consider are continuous. Furthermore, we examine the role of the strong Cantor topology in relation to indexed monoids, both with and without units, and display them as topological monoids in the strong Cantor topology. The totality of our results gives considerable support to the view that the strong Cantor topology is the topology of formal methods.

The set of continuous maps of a space X to a space Y equipped with the topology of pointwise convergence will be denoted by Cp(X, Y). For Y = R the notation is abbreviated to Cp(X). We will be interested in connections between the topologies of spaces X and Cp(X). These connections have not been studied extensively. However, some progress has been made recently -interesting results have been obtained in problems of normality and final compactness of Cp(X) (e.g., see [i, 2]). We will study some problems associated with cardinalvalued invariants. All spaces considered are assumed to be completely regular. I. Separability Problems. The problem of separability of Cp(X) was first considered in [3~. The following result was obtained: TIIEOREM i. Cp(X) is separable if and only if X is condensed to a space of countable weight. The main result of the present note is: THEOREM 2. Cp(X) is hereditarily separable if and only if xn is hereditarily finally compact, n~-Ni==_ Proof. Necessity. Let Cp(X) be hereditarily separable. Statement i. Suppose that ~C~Cp(X) is hereditarily separable and separates points from closed sets (i.e., if x~F and F is closed, then there is f ~ ~ such that f(x) = 1 and f(F) ~ 0). Then X is hereditarily finally compact~ Indeed, let ? = {H} be a family of sets open in Xo For each point x ~ (here, ~ = [J {H: H~---_?} is the body of y), we choose H~:? and ]~__~ such that fx(x) = 1 and f~(x \ H~)~0 There exists a countable set B~=-~ such that {/2: x=_B} is dense in {/~: x~}. Then %% = U {H~: x~-B} Indeed, if x:~_~, then there exists g~B subject to condition [/x (x) --f~ (x) [

A uniform topology τ M {\tau _M} induced by a set M M of finite measures on a quantum logic L L is studied. If m m is a valuation on L L , the topology τ m {\tau _m} induced by { m } \{ m\} is equivalent to the topology induced by the pseudometric ρ ( a , b ) = m ( a Δ b ) \rho (a,b) = m(a\Delta b) . If the set M M of measures is large enough, the topology τ M {\tau _M} reflects in some sense the structure of L L : if L L is a continuous geometry and the measures are totally additive, τ M {\tau _M} is weaker than the order topology τ o {\tau _o} on L L . If L L is atomic, τ M {\tau _M} is stronger than τ o {\tau _o} . On a separable Hilbert space logic, τ M {\tau _M} coincides with the discrete topology. Some cases are found in which τ M = τ o {\tau _M} = {\tau _o} .

This paper gives improvements and new results along the lines of the work of Rockafellar, Olech, and Sainte-Beuve. It is concerned with closure of sets and lower semicontinuity of functionals relative to the weak topologies on L1 or on the space M of measures.

The main result of this paper is that any proper collared embedding of R n − 1 {R^{n - 1}} in R n {R^n} can be extended to a homeomorphism of R n {R^n} such that the extension depends continuously on the original embedding in a stronger sense than previously known. Analogous results are proved for proper embeddings of R k {R^k} in R n {R^n} (with the usual homotopy conditions when k = n − 2 k = n - 2 ). An alternative proof of the usual compact form of the weak canonical Schoenflies theorem is also obtained.

Section 1 is a brief introduction. Section 2 contains the basic definitions of quasimanuals, weights, and operational logics. The linear spaceW of all weights on a quasimanualA is introduced and given a norm.W with this norm is seen to be a Banach space. The subspaceV ofW generated by the positive cone ofW is given the base norm and is also shown to be an Archimedian ordered Banach space with an additive norm. In Section 3 normal linear functionals onV * are defined in analogy with normal linear functionals onw * algebras. The spaceV is shown to be the set of normal functionals onV * and we showV to be the unique partially ordered Banach space with a closed generating cone which is predual toV *. Next, weakly compact subsets ofW are characterized in terms of eventwise convergence. This is the Hahn-Vitali-Saks theorem of classical measure theory in this noncommutative setting; several weak compactness results are drawn from this and compared with their classical counterparts. Section 4 introduces the ultraweak topology forV * in analogy with the same for the trace class operators on Hubert space. Here the condition for a compact base for the cone ofV is examined and shown to be a poor and unnecessary hypothesis in many circumstances. Many connections with the existent literature are made and throughout the paper there are many examples and open questions.