Operator space structure and amenability for Figà-Talamanca-Herz algebras

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with , we use the operator space structure on to equip the Figà-Talamanca-Herz algebra A_p(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p?q?2 or 2?q?p and amenable G, the canonical inclusion A_q(G)⊂A_p(G) is completely bounded (with cb-norm at most , where is Grothendieck's constant). As an application, we show that G is amenable if and only if A_p(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.     

Primitive elements in rings of continuous functions

Let be a surjective continuous map between compact Hausdorff spaces. The map π induces, by composition, an injective morphism C(Y)→C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with algebraic properties of the ring extension C(Y)⊆C(X) in relation to topological properties of the map . We prove that if the extension C(Y)⊆C(X) has a primitive element, i.e., C(X)=C(Y)[f], then it is a finite extension and, consequently, the map π is locally injective. Moreover, for each primitive element f we consider the ideal and prove that, for a connected space Y, If is a principal ideal if and only if is a trivial covering.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

Holomorphic cohomology groups on compact Kähler complex spaces

Let (X,OX) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups Hq(X,C) carry a mixed Hodge structure. In particular they carry a weight filtration W−lHq(X,C) (l=0,…,q), and the graded quotients have a direct sum decomposition into holomorphic invariants as . Here we investigate the relationships between the above invariants for r=0 and the cohomology groups , where is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c₁(L) in has pure type (1,1).     

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.     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

Operator biflatness of the Fourier algebra and approximate indicators for subgroups

We investigate if, for a locally compact group G, the Fourier algebra A(G) is biflat in the sense of quantized Banach homology. A central rôle in our investigation is played by the notion of an approximate indicator of a closed subgroup of G: The Fourier algebra is operator biflat whenever the diagonal in G×G has an approximate indicator. Although we have been unable to settle the question of whether A(G) is always operator biflat, we show that, for , the diagonal in G×G fails to have an approximate indicator.     

Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H_n)_n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then is the quantum Gromov-Hausdorff limit of any sequence for the natural quantum metric structures and when the lifts of σ_n to converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C^*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.     

Edge-connectivity and edge-disjoint spanning trees

Given a graph G, for an integer c∈{2,…,|V(G)|}, define λ_c(G)=min{|X|:X⊆E(G),ω(G−X)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,

     

Aluthge iteration in semisimple Lie group

We extend, in the context of connected noncompact semisimple Lie group, two results of Antezana, Massey, and Stojanoff: Given 0<λ<1, (a) the limit points of the sequence are normal, and (b) , where ‖X‖ is the spectral norm and r(X) is the spectral radius of X∈C_n×n and Δ_λ(X) is the λ-Aluthge transform of X.     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)

     

A note on a theorem of Talagrand

We provide an elementary argument to show that if for a hemicompact kR-space X the space Cp(X) contains a subset S which separates the points of X and is dominated by irrationals, i.e. S is covered by a family of compact sets such that Kα⊂Kβ for α?β, then Cp(X) is also dominated by irrationals; consequently Cp(X) is K-analytic. This fact (which fails for non-hemicompact spaces X) extends an old result of Talagrand.     

Packing directed cycles efficiently

Let G be a simple digraph. The dicycle packing number of G, denoted ν_c(G), is the maximum size of a set of arc-disjoint directed cycles in G. Let G be a digraph with a nonnegative arc-weight function w. A function ψ from the set C of directed cycles in G to R₊ is a fractional dicycle packing of G if ∑_e∈C∈Cψ(C)?w(e) for each e∈E(G). The fractional dicycle packing number, denoted , is the maximum value of ∑_C∈Cψ(C) taken over all fractional dicycle packings ψ. In case w≡1 we denote the latter parameter by .Our main result is that where n=|V(G)|. Our proof is algorithmic and generates a set of arc-disjoint directed cycles whose size is at least ν_c(G)-o(n²) in randomized polynomial time. Since computing ν_c(G) is an NP-Hard problem, and since almost all digraphs have ν_c(G)=Θ(n²) our result is a FPTAS for computing ν_c(G) for almost all digraphs.The result uses as its main lemma a much more general result. Let F be any fixed family of oriented graphs. For an oriented graph G, let ν_F(G) denote the maximum number of arc-disjoint copies of elements of F that can be found in G, and let denote the fractional relaxation. Then, . This lemma uses the recently discovered directed regularity lemma as its main tool.It is well known that can be computed in polynomial time by considering the dual problem. We present a polynomial algorithm that finds an optimal fractional dicycle packing. Our algorithm consists of a solution to a simple linear program and some minor modifications, and avoids using the ellipsoid method. In fact, the algorithm shows that a maximum fractional dicycle packing with at most O(n²) dicycles receiving nonzero weight can be found in polynomial time.     

Weak barrelledness for C(X) spaces

Buchwalter and Schmets reconciled C_c(X) and C_p(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if -barrelled ⇔ ?^∞-barrelled for C_c(X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C_c(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C_p(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C_c(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C_c(X) is a df-space (Jarchow's 1981 question).     

Sequential properties of function spaces with the compact-open topology

The main results of the paper are:

(1)

If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;

(2)

For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X^′ is compact; and

(3)

All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

     

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S₀(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S₀(G) with an operator space structure. With this structure S₀(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L¹(G). We show that this operator space structure is consistent with the major functorial properties: (i) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map is completely surjective, if H is a closed subgroup; and (iii) is completely surjective, where N is a normal subgroup and . We also show that S₀(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.