Vol. 17, No. 1, 1966

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Centralizers and H∗-algebras

Charles N. Kellogg

Vol. 17 (1966), No. 1, 121–129

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, {ea}aD in the H-algebra such that {eax}aD and {xea}aD 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 self-adjoint 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.

Mathematical Subject Classification
Primary: 46.65
Secondary: 46.60
Received: 25 January 1965
Published: 1 April 1966
Charles N. Kellogg