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ISOMETRIES OF BANACH ALGEBRAS SATISFYING THE VON NEUMANN INEQUALITY
 

Summary: ISOMETRIES OF BANACH ALGEBRAS SATISFYING
THE VON NEUMANN INEQUALITY
Jonathan Arazy
(Mathematica Scandinavica Vol. 74 (1994), 137-151)
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
self-adjoint 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.

  

Source: Arazy, Jonathan - Department of Mathematics, University of Haifa

 

Collections: Mathematics