The Banach space E has the weakly compact approximation property (W.A.P.) if there is C < ∞ so that the identity map IE can be uniformly approximated on any weakly compact subset D ⊂ E by weakly compact operators V on E satisfying ‖V ‖ ≤ C. We show that the spaces N(`, `) of nuclear operators ` → ` have the W.A.P. for 1 < q ≤ p < ∞, but that the Hardy space H does not have the W.A.P.

Assuming Jensen's diamond principle (⋄) we construct for every natural number n>0 a compact Hausdorff space K such that whenever the Banach spaces C(K) and C(L) are isomorphic some L, then covering dimension of L is equal to n. The constructed separable connected, has few operators i.e. bounded linear operator T:C(K)→C(K) form T(f)=fg+S(f), where g∈C(K) S weakly compact.

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective corres...

The projective hull X̂ of a compact set X ⊂ P is an analogue of the classical polynomial hull of a set in C. In the special case that X ⊂ C ⊂ P, the affine part X̂ ∩C can be defined as the set of points x ∈ C for which there exists a constant Mx so that |p(x)| ≤ M x sup X |p| for all polynomials p of degree ≤ d, and any d ≥ 1. Let X̂(M) be the set of points x where Mx can be chosen ≤ M . Using an ...

The projective hull X̂ of a compact set X ⊂ P is an analogue of the classical polynomial hull of a set in C. In the special case that X ⊂ C ⊂ P, the affine part X̂ ∩C can be defined as the set of points x ∈ C for which there exists a constant Mx so that |p(x)| ≤ M x sup X |p| for all polynomials p of degree ≤ d, and any d ≥ 1. Let X̂(M) be the set of points x where Mx can be chosen ≤ M . Using an ...

For a locally compact group G we look at the group algebras C 0 (G) and C * r (G), and we let f ∈ C 0 (G) act on L 2 (G) by the multiplication operator M (f). We show among other things that the following properties are equivalent: 1. G has a compact open subgroup. 2. One of the C *-algebras has a dense multiplier Hopf *-subalg-ebra (which turns out to be unique). 3. There are non-zero elements...

We continue the study of multidimensional operator multipliers initiated in [12]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and...

A Tychonoff space X is called RG if the embedding of C(X) → C(Xδ) is an epimorphism of rings. Compact RG spaces are known and easily described. We study the pseudocompact RG spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The ma...

By proving that, if the quotient space Σ(X) of the connected components of the locally compact metric space (X, d) is compact, then the full group I(X, d) of isometries of X is closed in C(X, X) with respect to the pointwise convergence topology, i.e., that I(X, d) coincides in this case with its Ellis’ semigroup, we complete the proof of the following: Theorem (a): If Σ(X) is not compact, I(X,...

today each city due to natural limitations, facilities and planning policies has different growth pattern which according to its spread, has different consequences and results. although there is a wide range of urban growth patterns, but the important thing is not diversity and number of models but is the absence of rules and methods of transition from one to another pattern (form). so this art...