Hyperstonean cover and second dual extension

Zaharov, V. K.

doi:10.1007/bf01903626

Cited by 3 publications

(3 citation statements)

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“…The term 'hyper-Stonean cover' is used for our 'hyper-Stonean envelope' in [125], where some references to earlier works are given. In [125], there is a characterization of Ω in terms of certain 'Kelley ideals'.…”

Section: Second Duals Of Algebras Of Continuous Functionsmentioning

confidence: 99%

Second duals of measure algebras

Dales

Lau

Strauss

2011

Dissertationes Math.

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Let G be a locally compact group. We shall study the Banach algebras which are the group algebra L 1 (G) and the measure algebra M (G) on G, concentrating on their second dual algebras. As a preliminary we shall study the second dual C0(Ω) of the C * -algebra C0(Ω) for a locally compact space Ω, recognizing this space as C( Ω), where Ω is the hyper-Stonean envelope of Ω.We shall study the C * -algebra of B b (Ω) of bounded Borel functions on Ω, and we shall determine the exact cardinality of a variety of subsets of Ω that are associated with B b (Ω).We shall identify the second duals of the measure algebra (M (G), ) and the group algebra (L 1 (G), ) as the Banach algebras (M ( G), 2 ) and (M (Φ), 2 ), respectively, where 2 denotes the first Arens product and G and Φ are certain compact spaces, and we shall then describe many of the properties of these two algebras. In particular, we shall show that the hyper-Stonean envelope G determines the locally compact group G. We shall also show that ( G, 2 ) is a semigroup if and only if G is discrete, and we shall discuss in considerable detail the product of point masses in M ( G). Some important special cases will be considered.We shall show that the spectrum of the C * -algebra L ∞ (G) is determining for the left topological centre of L 1 (G) , and we shall discuss the topological centre of the algebra (M (G) , 2 ).

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Section: Second Duals Of Algebras Of Continuous Functionsmentioning

confidence: 99%

Second duals of measure algebras

Dales

Lau

Strauss

2011

Dissertationes Math.

View full text Add to dashboard Cite

show abstract

“…The author of the paper has proved that the a-preimage T ~--pT realizes the c-extension C ~ Z too. The concrete form of the realizing morphism was established by the author in [5,69,16,76] (see also [67]). …”

Section: If T Is a Baire Space Then The C-extension C ~ B Is Isomorpmentioning

confidence: 99%

“…According to the criterion from 1.3.2, this a-preimage exists and is a realizing asL-preimage of the crLextension u: C ~-+ C ~'. This preimage was examined in [32,34,35,39,46,48,49,76]. Since the hyperstonean preimage is defined via C t', Fremlin, in his review of the monograph [59], raised the problem of the ez.istence of a direct topological construction of the hyperstonean preimage T ~ hT out closed sets and, correspondingly, zero-sets of a space T. A function f: T ~ R will be called an A-function if for any n there exists a finite covering {A,,i I i} C A of the space T such that w(f, …”

Section: 32mentioning

confidence: 99%