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...

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...

A sequence ${T_n}_{n=1}^{infty}$ of bounded linear  operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the disk-scaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C}, | alpha | leq 1}cap M$ is dense in $M$. The goal of t...

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 20 = 21 , there exists a normal absolutely star countable discrete clos...

2 Product topology, Subspace topology, Closed sets, and Limit Points 6 2.1 The Product Topology on X × Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Subspace Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Closed Sets, Closure, Interior, and limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Sub...

We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position terms suitably defined Seshadri constants with respect to fixed ample divisor. Our proof builds on previous work Evertse and Ferretti, Corvaja Zannier, others, uses standard techniques from algebraic geometry such as notions positivity, blowing-ups direct image sheaves. As an application, we r...