The centroid of a weak*-closed inner ideal in a JBW*-triple

It is shown that there exists a ^*-homomorphism from the continuous centroid L^b (A){\cal L}^b (A) of a JBW^*-triple A onto the continuous centroid L^b (J){\cal L}^b (J) of an arbitrary weak^*-closed inner ideal J in A.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.     

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ^∞ (resp., noncommutative disc algebraA _n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂⁿ are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ^∞/J on Hilbert spaces whereJ is anyw ^*-closed, 2-sided ideal ofF ^∞, are obtained and used to construct aw ^*-continuous,F ^∞/J-functional calculus associated to row contractionsT=[T ₁,…,T _n] whenf(T₁, …, T_n)=0 for anyf∈J. Other properties of the dual algebraF ^∞/J are considered. The second author was partially supported by NSF DMS-9531954.     

A Banach-lattice version of the Josefson-Nissenzweig theorem

Let E, F be two Banach lattices with E order continuous. If F can be mapped positively onto E then the dual F^* contains a weak^* -null sequence of positive and norm-one elements (Theorem 1). This is a Banach-lattice version of the classical Josefson-Nissenzweig theorem. It is an immediate consequence of the dual characterization of order continuity: E is order continuous iff E is Dedekind complete and every norm-one and pairwise disjoint sequence in E^* is weak^*-null (Theorem 2).     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Weak generators of the algebral ∞ and the commutant of a model operator

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H^∞. It is known that ϕ+BH^∞ is a weak^* generator of the algebra H^∞/BH^∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak^* generator of the algebra l^∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B₁·…·B_N, where N ∈ℕ, and B₁, …, B_N are Blaschke products satisfying condition (C), there exists a function ϕ∈H^∞ such that ϕ(Λ) is a weak^* generator of the algebra l^∞, and ϕ+BH^∞ is not a weak^* generator of the algebra H^∞/BH^∞. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85. Translated by M. F. Gamal'.     

On the square root of a positive B( )-valued function

Let (Ω, , μ) be a measure space, a separable Banach space, and ^* the space of all bounded conjugate linear functionals on . Let f be a weak^* summable positive B( ^*)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B( )-valued function Q satisfying the relation Q^*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺( ^*)-valued measures, the concepts of weak^*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Cycles of Spatial Discretizations of Shadowing Dynamical Systems

Let H(B,α) be the JBW^*-algebra of elements of a continuous W^*-algebra B invariant under the ^*-anti-automorphism α of B of order two. Then the mapping I → I ∩ H(B, α) is an order isomorphism from the complete lattice of α-invariant weak^* closed inner ideals in B onto the complete lattice of weak^* closed inner ideals in H(B, α), every one of which is of the form eH(B, α) α(e) for some unique projection e in B with α-invariant central support. A corollary of this result completely characterizes the weak^* closed inner ideals in any continuous JBW^*-triple.     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

Translation-invariant subspaces of functions on the group of linear transformations on the real line

The characterization of right translation-invariant subspaces ofL _∞(G ^*), where , is studied. We introduce the class of multiplier functions which, in the semisimple case, play a role similar to that played by the exponentials for the real line. However, it is proved that multiplier functions ofG ^* with respect toR fail to characterize right translation-invariant subspaces ofL _∞(G ^*). That is, we construct a right translation-invariant, w^*-closed subspace ofL _∞(G ^*) which contains no multiplier function. This paper is a part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor H. Furstenberg, to whom the author wishes to express his thanks for his helpful guidance, and valuable remarks.     

Convex -closures versus convex norm-closures in dual Banach spaces

A subset Y of a dual Banach space X^* is said to have the property (P) if for every weak^*-compact subset H of Y. The purpose of this paper is to give a characterization of the property (P) for subsets of a dual Banach space X^*, and to study the behavior of the property (P) with respect to additions, unions, products, whether the closed linear hull has the property (P) when Y does, etc. We show that the property (P) is stable under all these operations in the class of weak^* -analytic subsets of X^*.     

Between closed sets andg-closed sets

Quite recently, by using semi-open (resp.α-open, preopen,β-open) sets in a topological space, the notions ofsg*-closed (resp.αg*-closed,pg*-closedβg*-closed) sets are indroduced and investigated in [8]. These subsets place between closed sets andg-closed sets due to Levine [5]. In this paper, we introduce the notion ofmg*-closed sets and obtain the unified theory for collections of subsets between closed sets andg-closed sets.     

James’ quasi-reflexive space is not isomorphic to any subspace of its dual

We prove that each non-reflexive subspace ofJ ^* contains a subspace isomorphic toJ ^* and complmented inJ ^*. Consequences are thatJ is not isomorphic to any subspace ofJ ^*, and that every reflexive subspace ofJ is contained in a complemented reflexive subspace ofJ.     

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Equations on the semidirect product of a finite semilattice by a finite commutative monoid

LetCom _t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx ^t =x ^t+q . The varietyCom _1,1 is the variety of finite semilattices also denoted byJ ₁. In this paper, we consider the product varietyJ ₁*Com _t,q generated by all semidirect products of the formM*N withM∈J ₁ andN∈Com _t,q . We give a complete sequence of equations forJ ₁*Com _t,q implying complete sequences of equations forJ ₁*(Com∩A),J ₁*(Com∩G) andJ ₁*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups. This material is based upon work supported by the National Science Foundation under Grants No. CCR-9101800 and CCR-9300738. Many thanks to the referee for his valuable comments and suggestions.     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

A criterion for polynomial growth of a-free two-peak posets of tame prinjective type *

It is well known from the results of L, A. Nazarova and A. G. Zavadskij [18], [19], see also [25, Chapter 15], that a poset J with one maximal element is of tame prinjective type and of polynomial growth if and only if J does not contain neither any of the Nazarova's hypercritical posets (1,1,1,1,1)^*, (1,1,1,2)^*,(2,2,3)^*, (1,3,4)^*,(W,5)^*,(1,2,6)^* nor the Nazarova-Zavadskij poset (NZ)^* (see Table 1 below). In the present paper we extend this result to a class of posets with two maximal elements. We show that Ã-free poset with two maximal elements is of tame representation type and of polynomial growth if and only if the Tits quadratic form q^s → Z (see (1.7) below) associated with J is weakly non-negative and J does not contain any of the six posets listed in Table 1 as a peak subposet.