A mapping T from a
Banach algebra X into itself shall be called a centralizer of X if x(Ty) = (Tx)y for all
x,y ∈ X. A bounded linear operator, T, in X shall be called a right [left] centralizer
if T(xy) = (Tx)y[T(xy) = x(Ty)]. We show that the space of centralizers forms a
closed commutative subalgebra of the bounded linear operators in X. The
intersection of the space of right centralizers with the space of left centralizers is
precisely the algebra of centralizers.
We show that the algebra of right [left] centralizers of an H^{∗}algebra is the
W^{∗}algebra generated by the left [right] multiplication operators and that the
commutant of the algebra of right [left] centralizers is the algebra of left [right]
centralizers. In order to do this, we construct a net, {e_{a}}_{a∈D} in the H^{∗}algebra such
that {e_{a}x}_{a∈D} and {xe_{a}^{∗}}_{a∈D} converge to x. We show that the algebra of
centralizers of a commutative H^{∗}algebra is the space of bounded functions
on a discrete set. Characterizations are given for compact and projection
centralizers.
We also study commutative H^{∗}algebras in which the irreducible selfadjoint
idempotents all have the same norm. We show that two such H^{∗}algebras are
topologically and algebraically equivalent if and only if they have the same Hilbert
space dimension.
