Skip to main content
1 of 3
Onur Oktay
  • 2.6k
  • 1
  • 7
  • 20

Contractive projections on operator algebras

This is a follow up on an earlier question.

Let $A$ be a non-separable unital (not necessarily self-adjoint) operator algebra, which is reflexive as a Banach space. Let $W$ be a $1$-complemented separable unital subalgebra, and $P:A\to A$ be a contractive projection onto $W$.

Question: Do we have $P(ux) = uP(x)$ for all $u\in W$, $x\in A$? If not, does there exist another $1$-complemented separable subalgebra with this property, which contains $W$?

PS: Could you please share general references in the comments about projections on Banach algebras, specifically about contractive projections whose range is a subalgebra?

Onur Oktay
  • 2.6k
  • 1
  • 7
  • 20