Gleason parts in bidual algebras
One of the methods of studying the Banach algebra H∞(Ω) is to view it as a subspace of the second dual A** of the algebra A=A(Ω). The first algebra includes all bounded analytic functions on a domain G⊂ Cd, while A(Ω) consists of elements in H∞(Ω) having continuous extensions to Ω and its spectrum is relatively easy to describe. The Arens multiplication in A** extends the product in A under the natural embedding and even in the more general setup of function algebras, the bidual algebra is Arens-regular. Our study of the behaviour of Gleason parts and their closures under the passage to the second dual space initiated in  is continued, showing the delicate nature of the matter.
Unfortunately, there was a gap in , spotted by H.G. Dales  whose study of the bidual of A=C(K) in  provided some hints in the cardinality of Gelfand closures of subsets of a compact space K. Consequently, most of the claims of Theorem 6 in  cannot hold in such generality. On the other hand, for the most interesting Gleason part related to the points of Ω we are able to obtain a number of positive results. Assuming some regularity, we show that the Gleason part in A** that corresponds to the domain Ω is not containing any other points than the evaluation functionals at (the canonical image) of Ω. This can be even continued to higer- order duals of A.
Also some results on bands of measures representing points of Ω and the closures of their canonical embeddings in the bidual spaces will be presented, including an alternative (new) proof of F. and M. Riesz Theorem.
The talk is based on joint work with Marek Kosiek.
- H. G. Dales, Private communication (IX 2017)
- H. G. Dales, F. K. Dashiell, Jr., A. T.-M. Lau, and D. Strauss, Banach Spaces of Continuous Functions as Dual Spaces, SpringerVerlag, New York, 2016.
- M. Kosiek, K. Rudol, Dual algebras and A – measures, Journ. of Function Spaces 2014, 1-8 http://dx.doi.org/10.1155/2014/364271