Mathematics Ph.D. Dissertations

Weak*-Closed Unitarily and Moebius Invariant Spaces of Bounded Measurable Functions on a Sphere

Date of Award

2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Alexander Izzo (Advisor)

Second Advisor

Kit Chan (Committee Member)

Third Advisor

Paul Moore (Other)

Fourth Advisor

Steven Seubert (Committee Member)

Abstract

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and Lp functions on a sphere, for 1≤p

Each set of pairs of nonnegative integers induces a weak*-closed unitarily invariant space, and conversely each weak*-closed unitarily invariant space is induced by such a set. For algebras, we show that a set of pairs of nonnegative integers induces a weak*-closed unitarily invariant algebra of L∞ functions on a sphere if and only if the set induces a closed unitarily invariant algebra of continuous functions on a sphere. Thus, the same criterion which Rudin formulates in his 1979 paper to characterize the sets which induce closed unitarily invariant algebras of continuous functions on a sphere for dimension at least 3 also serves to characterize the sets which induce weak*-closed unitarily invariant algebras of L∞ functions on a sphere for dimension at least 3. Further, the same exceptions which arise in dimension 2 for continuous functions also arise in dimension 2 for L∞ functions.

Finally, we determine the weak*-closed Möbius invariant spaces of L∞ functions on a sphere, of which there are six: the null space, the constant functions, the space of almost everywhere radial limits of holomorphic functions on the ball, the space of conjugates of members of the previous space, the weak*-closure of the algebraic sum of the previous two spaces, and the space of L∞ functions. Of these spaces, all are algebras except the fifth, which is not an algebra if and only if the dimension of the sphere is greater than 1.

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