Dual algebras and A-measures. Marek Kosiek Joint work with Krzysztof Rudol Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Motivation: Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Motivation: A - measures problem Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Motivation: A - measures problem the problem for which G ⊂ C n the algebra H ∞ ( G ) is a dual algebra Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Motivation: A - measures problem the problem for which G ⊂ C n the algebra H ∞ ( G ) is a dual algebra the application of dual algebras in functional calculus for bounded operators in Hilbert spaces Marek Kosiek Dual algebras and A-measures.
A - an arbitrary function algebra The main subject of our investigation: Properties of the spectrum of A ∗∗ . Motivation: A - measures problem the problem for which G ⊂ C n the algebra H ∞ ( G ) is a dual algebra the application of dual algebras in functional calculus for bounded operators in Hilbert spaces connections with the Corona problem Marek Kosiek Dual algebras and A-measures.
As an application of our main result we have obtained: Marek Kosiek Dual algebras and A-measures.
As an application of our main result we have obtained: a general positive solution for A - measures problem Marek Kosiek Dual algebras and A-measures.
As an application of our main result we have obtained: a general positive solution for A - measures problem the duality of H ∞ ( G ) algebra for some classes of bounded domains G ⊂ C n Marek Kosiek Dual algebras and A-measures.
Definition A is a function algebra on a compact set X iff A ⊂ C ( X ) , A contains constants and separates the points of X Marek Kosiek Dual algebras and A-measures.
Definition A is a function algebra on a compact set X iff A ⊂ C ( X ) , A contains constants and separates the points of X Let φ, ψ ∈ σ ( A ) df φ ∼ ψ ⇐ ⇒ � φ − ψ � < 2 Marek Kosiek Dual algebras and A-measures.
Definition A is a function algebra on a compact set X iff A ⊂ C ( X ) , A contains constants and separates the points of X Let φ, ψ ∈ σ ( A ) df φ ∼ ψ ⇐ ⇒ � φ − ψ � < 2 Definition The equivalence classes in the above equivalence relation are called Gleason parts of A . Marek Kosiek Dual algebras and A-measures.
Definition A is a function algebra on a compact set X iff A ⊂ C ( X ) , A contains constants and separates the points of X Let φ, ψ ∈ σ ( A ) df φ ∼ ψ ⇐ ⇒ � φ − ψ � < 2 Definition The equivalence classes in the above equivalence relation are called Gleason parts of A . We assume σ ( A ) = X . Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M every measure µ ∈ M ( X ) has a unique Lebesque decomposition µ = µ M + µ s where µ M ∈ M and µ s is singular to each measure in M Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M every measure µ ∈ M ( X ) has a unique Lebesque decomposition µ = µ M + µ s where µ M ∈ M and µ s is singular to each measure in M M is a reducing band (with respect to A) if µ ∈ A ⊥ = ⇒ µ M ∈ A ⊥ Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M every measure µ ∈ M ( X ) has a unique Lebesque decomposition µ = µ M + µ s where µ M ∈ M and µ s is singular to each measure in M M is a reducing band (with respect to A) if µ ∈ A ⊥ = ⇒ µ M ∈ A ⊥ ν is a representing measure for x ∈ X = σ ( A ) if � f ( x ) = f d ν for f ∈ A Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M every measure µ ∈ M ( X ) has a unique Lebesque decomposition µ = µ M + µ s where µ M ∈ M and µ s is singular to each measure in M M is a reducing band (with respect to A) if µ ∈ A ⊥ = ⇒ µ M ∈ A ⊥ ν is a representing measure for x ∈ X = σ ( A ) if � f ( x ) = f d ν for f ∈ A for G ⊂ X we denote by M G the band generated by G i.e. the smallest band containing all measures representing for points in G Marek Kosiek Dual algebras and A-measures.
M ⊂ M ( X ) = C ( X ) ∗ is a band if it is a closed subspace and µ ∈ M , ν ≪ | µ | = ⇒ ν ∈ M every measure µ ∈ M ( X ) has a unique Lebesque decomposition µ = µ M + µ s where µ M ∈ M and µ s is singular to each measure in M M is a reducing band (with respect to A) if µ ∈ A ⊥ = ⇒ µ M ∈ A ⊥ ν is a representing measure for x ∈ X = σ ( A ) if � f ( x ) = f d ν for f ∈ A for G ⊂ X we denote by M G the band generated by G i.e. the smallest band containing all measures representing for points in G if G is a Gleason part then M G is a reducing band Marek Kosiek Dual algebras and A-measures.
( C ( X ) ∗ ) ∗ = M ( X ) ∗ ≈ C ( Y ) for some hyperstonean compact space Y Marek Kosiek Dual algebras and A-measures.
( C ( X ) ∗ ) ∗ = M ( X ) ∗ ≈ C ( Y ) for some hyperstonean compact space Y each f ∈ C ( X ) can be treated as a functional on M ( X ) and consequently as an element of C ( Y ) by the formula � � f , µ � = f d µ for µ ∈ M ( X ) Marek Kosiek Dual algebras and A-measures.
( C ( X ) ∗ ) ∗ = M ( X ) ∗ ≈ C ( Y ) for some hyperstonean compact space Y each f ∈ C ( X ) can be treated as a functional on M ( X ) and consequently as an element of C ( Y ) by the formula � � f , µ � = f d µ for µ ∈ M ( X ) for µ ∈ M ( X ) there is a unique measure µ ∈ M ( Y ) = C ( Y ) ∗ such that � F , µ � = � ˜ F d ˜ µ for all F ∈ C ( Y ) Marek Kosiek Dual algebras and A-measures.
Theorem ws of G If G is a Gleason part of A then the weak-star closure G in Y is a closed-open subset of Y . Moreover ws = X \ G ws = M ( G ws , ws ) s = ( M s ws , ws ) , Y \ G ( M G G ) M G ws is a reducing band for A ∗∗ . and M G Marek Kosiek Dual algebras and A-measures.
Corollary There exists a characteristic function F 0 ∈ A ∗∗ vanishing exactly ws and the projection associated with the on Y \ G ws is exactly the ws + M s decomposition M ( Y ) = M G G multiplication by F 0 . Marek Kosiek Dual algebras and A-measures.
Corollary There exists a characteristic function F 0 ∈ A ∗∗ vanishing exactly ws and the projection associated with the on Y \ G ws is exactly the ws + M s decomposition M ( Y ) = M G G multiplication by F 0 . Corollary If G is a Gleason part of a function algebra A , x ∈ G and µ x is any its representing measure, then µ x is concentrated on the weak-star closure of G . Marek Kosiek Dual algebras and A-measures.
G - a Gleason part of A Marek Kosiek Dual algebras and A-measures.
G - a Gleason part of A H ∞ ( M G ) - the weak-star closure of A in M ∗ G Marek Kosiek Dual algebras and A-measures.
G - a Gleason part of A H ∞ ( M G ) - the weak-star closure of A in M ∗ G by the definition of H ∞ ( M G ) , the values of its every element are uniquely defined on each x ∈ G Marek Kosiek Dual algebras and A-measures.
G - a Gleason part of A H ∞ ( M G ) - the weak-star closure of A in M ∗ G by the definition of H ∞ ( M G ) , the values of its every element are uniquely defined on each x ∈ G Proposition H ∞ ( M G ) is isometrically isomorphic to A ∗∗ / M ⊥ G ∩ A ∗∗ Marek Kosiek Dual algebras and A-measures.
G - a Gleason part of A H ∞ ( M G ) - the weak-star closure of A in M ∗ G by the definition of H ∞ ( M G ) , the values of its every element are uniquely defined on each x ∈ G Proposition H ∞ ( M G ) is isometrically isomorphic to A ∗∗ / M ⊥ G ∩ A ∗∗ Corollary G is a subset of the spectrum of H ∞ ( M G ) Marek Kosiek Dual algebras and A-measures.
Theorem If G is a Gleason part of A , then H ∞ ( M G ) satisfies the domination condition: f ∈ H ∞ ( M G ) � f � = sup | f ( x ) | for any x ∈ G Marek Kosiek Dual algebras and A-measures.
Theorem If G is a Gleason part of A , then H ∞ ( M G ) satisfies the domination condition: f ∈ H ∞ ( M G ) � f � = sup | f ( x ) | for any x ∈ G Proposition The band M G is equal to the norm closed linear span of all representing measures for points in G , taken in the quotient space M ( X ) / A ⊥ . Marek Kosiek Dual algebras and A-measures.
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