 
Summary: ISOMETRIES OF BANACH ALGEBRAS SATISFYING
THE VON NEUMANN INEQUALITY
Jonathan Arazy
(Mathematica Scandinavica Vol. 74 (1994), 137151)
Introduction
A famous classical result of R.Kadison [K] says that every isometry of one C algebra
onto another is given by a Jordan isomorphism followed by a unitary multiplication. This
result is generalized in the recent works [AS], [MT1] and [MT2] on isometries of certain non
selfadjoint algebras of operators on Hilbert space. In the context of these works it is even
possible to describe explicitly the Jordan isomorphisms. In this paper we generalize Kadi
son's theorem even further to the context of Banach algebras satisfying the von Neumann
inequality.
In what follows Z denotes a complex, unital Banach algebra (i.e. Z has a unit e and
k e k= 1) with an open unit ball D. The term "Banach algebra" will always mean "complex,
unital Banach algebra". We say that the von Neumann inequality hold in Z if
k f(z) kk f k 1 := maxfj f() j; j j= 1g
for every z 2 D and every polynomial f. It is well known that the von Neumann inequal
ity holds in B(H), the algebra of all bounded operators on the Hilbert space H, see [N],
[FSN, Chapter I.8]. Therefore, it holds in every subalgebra of a C  algebra. The following
generalization of Kadison's theorem is our main result.
