Search

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

We prove that two automorphisms of L??-spaces are conjugate if and only if certain related operator algebras are algebraically isomorphic. This extends a result of W. Arveson by dropping the assumptions that the automorphisms are ergodic and measure-preserving. W. B. Arveson [1] first looked at the conjugacy problem for automorphisms of measure spaces in terms of operator algebras. Suppose (X, ,u) is a finite measure space and a is an ergodic measure-preserving automorphism of L? (,u) . Arveson [1] showed that the Banach algebra of operators on L2(X, u) generated by the unitary operator Ua of composition with a, together with the multiplications by L? (X, ,u)-functions, classifies a in the sense that two such automorphisms are conjugate if and only if the associated algebras are unitarily equivalent. Later, Arveson and K. B. Josephson [2] extended this result by showing that the algebras need only be isomorphic. This was proved not in the setting of measure-preserving automorphisms but of homeomorphisms on locally compact spaces. The Arveson-Josephson result was later extended by J. Peters [4] using semi-crossed products. In [3] the present authors associated with each homeomorphism on a compact Hausdorff space a family of algebras called conjugacy algebras. It was then proved that two homeomorphisms are conjugate if and only if some conjugacy algebra for the first is isomorphic to some conjugacy algebra for the second. The algebras considered in [1, 2], and [4] are conjugacy algebras, so the results in [3] include these. In this note, we return to the automorphisms of Arveson's original paper [1]. We show that two such automorphisms are conjugate if the corresponding algebras are isomorphic, and we do not need to assume that the automorphisms Received by the editors June 24, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D05, 47D25; Secondary 46L40, 47D30.

A family of algebras is associated with each continuous map on a compact Hausdorff space. These algebras are called conjugacy algebras. It is shown that two continuous maps are conjugate if and only if some conjugacy algebra of one is isomorphic to some conjugacy for the other. Numerous examples of conjugacy algebras are provided.