Disjointness preserving operators between little Lipschitz algebras

Given a real number α∈(0,1) and a metric space (X,d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that

     

Zero product preserving maps on Banach algebras of Lipschitz functions

Let (K,d) be a non-empty, compact metric space and α∈]0,1[. Let A be either lipα(K) or Lipα(K) and let B be a commutative unital Banach algebra. We show that every continuous linear map T:A→B with the property that T(f)T(g)=0 whenever f,g∈A are such that fg=0 is of the form T=wΦ for some invertible element w in B and some continuous epimorphism Φ:A→B.     

Maximal ideals of disjointness preserving operators

Let L and M be vector lattices with M Dedekind complete, and let Lr(L,M) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in Lr(L,M) (briefly, maximal δ-ideals of Lr(L,M)) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this ‘new’ structure. In this regard, various standard facts on orthomorphisms are extended to maximal δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of Lr(L,M) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Polar decomposition of order bounded disjointness preserving operators

We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).

     

On Fourier algebra homomorphisms

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows

     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T(N) acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T(N)-bimodule containing the nest algebra. They also prove that Hcbn(A, M)(?)Hcbn(A, A) for all n≥1 and a CSL algebra A with an ultraweakly closed A-bimodule M containing A.     

Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of A_p(G) algebras split strongly. Furthermore, each extension of A_p(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GL_n(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of A_p(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of A_p(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of B_p(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between A_p(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.     

Weakly compact composition operators between algebras of bounded analytic functions

We prove a characterization (up to the approximation property) of weakly compact composition operators in terms of their inducing analytic maps .

     

保不交算子值域的一些性质

研究了保不交算子值域的性质,建立了保不交算子值域为Riesz子空间的一个刻画;又讨论了主理想和主带在保不交算子作用后的象的性质,一些相关结果也得以讨论.     

Fourier algebras on homogeneous spaces

Spectral synthesis and operator synthesis on a homogeneous space G/K, where K is a compact subgroup of a locally compact group G, are studied. Injection theorem for sets of spectral synthesis for A(G/K) is proved, extending the classical result of Reiter and more recent results of Kaniuth–Lau, Parthasarathy–Prakash and others. A simple direct image theorem for spectral synthesis is proved and an extension of the subgroup theorem and an alternate proof of the injection theorem are obtained as consequences. The relation between synthesis in the Fourier algebra A(G/K) and an appropriate Varopoulos algebra is obtained, subsuming earlier results of Varopoulos, Spronk–Turowska and Parthasarathy–Prakash. Study of relations between spectral synthesis and operator synthesis pioneered by Arveson and carried forward recently by Shulman–Turowska, Parthasarathy–Prakash and Ludwig–Turowska is undertaken on homogeneous spaces. Operator space methods are needed for this study, and more specifically, a characterisation of completely bounded multipliers on A(G/K) as the invariant part of a suitable weak^? Haagerup tensor product (or the space of Schur multipliers) is given and is used for this study.     

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra L ^∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ^∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ^∞(G) by to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to . We study the corresponding space of “σ-harmonic operators”, i.e., fixed points in under the action of σ. We show, under mild conditions on either σ or G, that is in fact a von Neumann subalgebra of . Our investigation of relies, in particular, on a notion of support for an arbitrary operator in that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to via ideals in , where denotes the trace class operators on L ²(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis. M. Neufang was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach. V. Runde was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach.     

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator on a normed lattice such that is not disjointness preserving.

     

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π:A(G)→B(H) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].     

Bounded and completely bounded local derivations from certain commutative semisimple Banach algebras

We show that for a locally compact group , every completely bounded local derivation from the Fourier algebra into a symmetric operator -module or the operator dual of an essential -bimodule is a derivation. Moreover, for amenable we show that the result is true for all operator -bimodules. In particular, we derive a new proof to a result of N. Spronk that is always operator weakly amenable.

     

Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets

Let be a uniform algebra, and let be a self-map of the spectrum of that induces a composition operator on . The object of this paper is to relate the notion of ``hyperbolic boundedness' introduced by the authors in 2004 to the essential spectrum of . It is shown that the essential spectral radius of is strictly less than if and only if the image of under some iterate of is hyperbolically bounded. The set of composition operators is partitioned into ``hyperbolic vicinities" that are clopen with respect to the essential operator norm. This partition is related to the analogous partition with respect to the uniform operator norm.

     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.