Two dimensional commutative Banach algebras and von Neumann inequality Original Research Article

We show the following: Let image be a two dimensional commutative Banach algebra with identity. If image satisfies image whenever f is a polynomial satisfying f(z)less-than-or-equals, slant1(zless-than-or-equals, slant1), then image is isometric to a subalgebra of the algebra B(H) of all bounded linear operators on some Hilbert space H, and image satisfies image whenever f is a polynomial in n variables satisfying f(z1,…,zn)less-than-or-equals, slant1 (zkless-than-or-equals, slant1,k=1,…,n), for all n.