PFA and complemented subspaces of ℓ∞/c0

Weak*-closed invariant subspaces and ideals of semigroup algebras on foundation semigroups

Let S be a locally compact foundation semigroup with identity and                          be its semigroup algebra. Let X be a weak*-closed left translation invariant subspace of    In this paper, we prove that  X  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. We also prove that a foundation semigroup with identity S is left amenab...

Learning with $\ell^{0}$-Graph: $\ell^{0}$-Induced Sparse Subspace Clustering

ℓ 1-graph [19, 4], a sparse graph built by reconstructing each datum with all the other data using sparse representation , has been demonstrated to be effective in clustering high dimensional data and recovering independent subspaces from which the data are drawn. It is well known that ℓ 1-norm used in ℓ 1-graph is a convex relaxation of ℓ 0-norm for enforcing the sparsity. In order to handle g...

A New Class of Banach Sequence Spaces

Banach and operator space structure of C ∗ - algebras

Introduction A C∗-algebra is often thought of as the non-commutative generalization of a C(K)-space, i.e. the space of continuous functions on some locally compact Hausdorff space, vanishing at infinity. We go one step further, for we seek to compare the Banach space properties of C∗-algebras and their naturally complemented subspaces, with those of C(K)-spaces. (For a recent survey on C(K) spa...

Closed Ideals of Operators on and Complemented Subspaces of Banach Spaces of Functions with Countable Support

Let λ be an infinite cardinal number and let `∞(λ) denote the subspace of `∞(λ) consisting of all functions which assume at most countably many non zero values. We classify all infinite dimensional complemented subspaces of `∞(λ), proving that they are isomorphic to `∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on `∞(λ) or `∞(λ) has the u...