In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.

Let H be a unitary (finite dimensional Hilbert) space. It is known [1] that a self-adjoint operator on H, A = A* ≥ 0, is a sum of n orthogonal projections for some n ∈ N if and only if 1) tr A ∈ N∪{0}, 2) tr A ≥ dim Im A ...

In the space of irreducible unitary representations of a linear group over an algebra of type Ā₂ an open dense subset of representations in the general position is singled out. This set is identified, up to simple ...

The problem of unitarization of representations of algebras generated by idempotents with linear relations is studied. Construction of non-unitarizable representations for some subintervals of continuous spectrum is ...