Finite dimensional Q-algebras and von Neumann inequality
by
Takanori Yamamoto
Hokkai-Gakuen University
Coauthors: Takahiko Nakazi (Hokkaido University)

Let B be a finite dimensional commutative Banach algebra with identity. Let B(H) denote the algebra of all bounded linear operators on some Hilbert space H. It is well known that if dim B >= 5, then the following statement is false. Ïf B satisfies the von Neumann inequality: T in B, ||T|| <= 1   ===>   ||f(T)|| <= 1 whenever f is a polynomial satisfying |f(z)| <= 1   (|z| <= 1), then B is isometric to a subalgebra of B(H), and B satisfies the von Neumann inequality of n variables: T_k in B,   ||T_k|| <= 1   (k=1, ..., n)   ===>   ||f(T₁, ..., T_n)|| <= 1 whenever f is a polynomial in n variables satisfying |f(z₁, ..., z_n)| <= 1     (|z_k| <= 1,  k=1, ..., n), for all n." We shall prove that if dim B = 2, then this statement is true. It seems difficult when dim B = 3, 4. We shall consider the case when dim B = 3.

Date received: May 23, 2002