Geometrical properties of Banach spaces and the distribution of the norm for a stable measure
Dans une algèbre de Banach et dans deux cas particuliers, nous montrons la continuité du centre du plus petit disque contenant le spectre. Pour a ∈ , on donne une condition nécessaire et suffisante pour avoir où d(a) est la distance de a aux scalaires et le rayon du plus petit disque contenant K qui représente le spectre ou le domaine numérique algébrique de a. Dans un espace de Hilbert complexe, K peut représenter certains types de spectres ou de domaines numériques de a.
A separable Banach space X contains isomorphically if and only if X has a bounded fundamental total -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total -biorthogonal system.
Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections determined by the different involutions induced by positive invertible elements a ∈ A. The maps sending p to the unique with the same range as p and sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that...
It is proved that if X is a rotund Banach space and M is a closed and proximinal subspace of X, then the quotient space X/M is also rotund. It is also shown that if Φ does not satisfy the δ₂-condition, then is not proximinal in and the quotient space is not rotund (even if is rotund). Weakly nearly uniform convexity and weakly uniform Kadec-Klee property are introduced and it is proved that a Banach space X is weakly nearly uniformly convex if and only if it is reflexive and it has the weakly...
A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of then X contains a copy of c₀. Moreover,...
Let be a unital Banach algebra over , and suppose that the nonzero spectral values of and are discrete sets which cluster at , if anywhere. We develop a plane geometric formula for the spectral semidistance of and which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that and are quasinilpotent equivalent if...