An estimate of the norms of inner derivation in some operator algebras (Q582541)

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. Given \(A\in B(H)\) denote by \(\Delta_ A\) the inner derivation \(B\to AB-BA\) of B(H). For a von Neumann algebra \(R\subset B(H)\) let \(\| \Delta_ A|_ R\|\) be the norm of the restriction of \(\Delta_ A\) onto R. The authors prove that \[ \| \Delta_ A|_ R\| \leq \pi \sup \{\| (I-P)AP\|:\quad P\in lat(R')\} \] where \(lat(R')\) is the lattice of all \(R'\)-invariant projections. When R is the algebra of all Toeplitz operators, the seminorms dist(A,R) and \(\| \Delta_ A|_ R\|\) are proved to e equivalent.