Beyond fractality Steffen Roch Technische Universit¨ at Darmstadt, Germany 1 / 21
Approximation sequences and stability Let A ∈ L ( l 2 ) and P n : l 2 → l 2 , ( x n ) n ≥ 0 �→ ( x 0 , . . . , x n − 1 , 0 , 0 , . . . ) . To solve an operator equation Au = f numerically by the finite sections discretization (FSD), consider the sequence of the equations ( P n A | im P n ) u n = P n f, n = 1 , 2 , . . . 2 / 21
Approximation sequences and stability Let A ∈ L ( l 2 ) and P n : l 2 → l 2 , ( x n ) n ≥ 0 �→ ( x 0 , . . . , x n − 1 , 0 , 0 , . . . ) . To solve an operator equation Au = f numerically by the finite sections discretization (FSD), consider the sequence of the equations ( P n A | im P n ) u n = P n f, n = 1 , 2 , . . . The sequence ( P n A | im P n ) is stable if there is an n 0 such that the P n A | im P n are invertible for n ≥ n 0 and their inverses are uniformly bounded. 2 / 21
Algebras of approximation sequences Let F stand for the set of all bounded sequences ( A n ) of operators A n : im P n → im P n . Provided with the operations ( A n ) + ( B n ) = ( A n + B n ) , ( A n )( B n ) = ( A n B n ) , ( A n ) ∗ = ( A ∗ n ) and the supremum norm, F becomes a C ∗ -algebra, and G = { ( A n ) : � A n � → 0 } is a closed ideal of F . 3 / 21
Algebras of approximation sequences Let F stand for the set of all bounded sequences ( A n ) of operators A n : im P n → im P n . Provided with the operations ( A n ) + ( B n ) = ( A n + B n ) , ( A n )( B n ) = ( A n B n ) , ( A n ) ∗ = ( A ∗ n ) and the supremum norm, F becomes a C ∗ -algebra, and G = { ( A n ) : � A n � → 0 } is a closed ideal of F . Stability theorem (Kozak) A sequence ( A n ) ∈ F is stable if and only if the coset ( A n ) + G is invertible in the quotient algebra F / G . 3 / 21
Example: The algebra of the FSD for Toeplitz operators For a ∈ C ( T ) , with k th Fourier coefficient a k , the Toeplitz operator T ( a ) ∈ L ( l 2 ) is given by its matrix representation ( a i − j ) i,j ≥ 0 . The Toeplitz algebra T ( C ) is the smallest closed subalgebra of L ( l 2 ) which contains all Toeplitz operators T ( a ) with a ∈ C ( T ) . The algebra of the FSD for Toeplitz operators S ( T ( C )) is the smallest closed subalgebra of F , the algebra of all bounded sequences ( A n ) with A n : im P n → im P n , which contains all sequences ( P n T ( a ) P n ) with a ∈ C ( T ) . 4 / 21
The algebra S ( T ( C )) of the FSD for Toeplitz operators Theorem (B¨ ottcher, Silbermann 1983) ( a ) S ( T ( C )) consists exactly of all sequences ( A n ) where A n = P n T ( a ) P n + P n KP n + R n LR n + G n with a ∈ C ( T ) , K, L compact and ( G n ) ∈ G . This representation is unique. ( b ) A sequence A = ( A n ) ∈ S ( T ( C )) is stable (i.e., A / G is invertible) if and only if W ( A ) := s-lim A n P n and � W ( A ) := s-lim R n A n R n are invertible. Here, R n : l 2 → l 2 , ( x n ) n ≥ 0 �→ ( x n − 1 , . . . , x 0 , 0 , 0 , . . . ) . 5 / 21
Fractal algebras Given η : N → N strictly increasing, let F η := { ( A η ( n ) ) : ( A n ) ∈ F} the restricted algebra, R η : F → F η , ( A n ) �→ ( A η ( n ) ) the restriction mapping. 6 / 21
Fractal algebras Given η : N → N strictly increasing, let F η := { ( A η ( n ) ) : ( A n ) ∈ F} the restricted algebra, R η : F → F η , ( A n ) �→ ( A η ( n ) ) the restriction mapping. Definition of a fractal algebra Let A be a C ∗ -subalgebra of F . ( a ) a homomorphism W : A → B is fractal if ∀ η ∃ W η : A η → B such that W = W η R η | A . ( b ) the algebra A is fractal if the canonical homomorphism π : A → A / ( A ∩ G ) is fractal. 6 / 21
Fractal algebras Given η : N → N strictly increasing, let F η := { ( A η ( n ) ) : ( A n ) ∈ F} the restricted algebra, R η : F → F η , ( A n ) �→ ( A η ( n ) ) the restriction mapping. Definition of a fractal algebra Let A be a C ∗ -subalgebra of F . ( a ) a homomorphism W : A → B is fractal if ∀ η ∃ W η : A η → B such that W = W η R η | A . ( b ) the algebra A is fractal if the canonical homomorphism π : A → A / ( A ∩ G ) is fractal. Example: S ( T ( C )) is fractal. 6 / 21
Properties of sequences in fractal algebras Let ( A n ) be a sequence in a fractal subalgebra of F . Then lim � A n � exists and is equal to � ( A n ) + G� , if ( A n ) = ( A n ) ∗ , then lim spec ( A n ) = spec (( A n ) + G ) , lim spec ε ( A n ) = spec ε (( A n ) + G ) , ( A ∩ K ) / G is a dual algebra, and more... 7 / 21
Properties of sequences in fractal algebras Let ( A n ) be a sequence in a fractal subalgebra of F . Then lim � A n � exists and is equal to � ( A n ) + G� , if ( A n ) = ( A n ) ∗ , then lim spec ( A n ) = spec (( A n ) + G ) , lim spec ε ( A n ) = spec ε (( A n ) + G ) , ( A ∩ K ) / G is a dual algebra, and more... Theorem A C ∗ -subalgebra A of F is fractal if and only if the limit lim � A n � exists for every sequence ( A n ) ∈ A . 7 / 21
The fractal restriction theorem For every separable C ∗ -subalgebra A of F , there is a strictly increasing η such that A η is fractal. (Proof: Diagonal argument.) 8 / 21
The fractal exhaustion theorem For every separable C ∗ -subalgebra A of F , there exists a (finite or infinite) number of strictly increasing sequences η 1 , η 2 , . . . with η i ( N ) ∩ η j ( N ) = ∅ for i � = j and ∪ i η i ( N ) = N such that every restriction A η i is fractal. (Proof: Repeated use of the fractal restriction theorem.) 9 / 21
The fractal exhaustion theorem For every separable C ∗ -subalgebra A of F , there exists a (finite or infinite) number of strictly increasing sequences η 1 , η 2 , . . . with η i ( N ) ∩ η j ( N ) = ∅ for i � = j and ∪ i η i ( N ) = N such that every restriction A η i is fractal. (Proof: Repeated use of the fractal restriction theorem.) We call A piecewise fractal if the number of restrictions in the fractal exhaustion theorem is finite. A is quasifractal if every restriction of A has a fractal restriction. 9 / 21
Example: Full FSD for Block Toeplitz operators Consider Toeplitz operators T ( a ) with a : T → C N × N continuous. Let S ( T ( C N × N )) denote the related algebra of the (full) FSD. Theorem ( a ) S ( T ( C N × N )) consists exactly of all sequences ( A n ) where A n = P n T ( a ) P n + P n KP n + R n L κ ( n ) R n + G n with a ∈ C ( T ) N × N , K, L i compact, ( G n ) ∈ G , and κ ( n ) is the remainder of n mod N . ( b ) A sequence A = ( A n ) ∈ S ( T ( C )) is stable if and only if W ( A ) := s-lim A n P n and � W i ( A ) := s-lim R nN + i A nN + i R nN + i are invertible. 10 / 21
Example: Full FSD for Block Toeplitz operators Consider Toeplitz operators T ( a ) with a : T → C N × N continuous. Let S ( T ( C N × N )) denote the related algebra of the (full) FSD. Theorem ( a ) S ( T ( C N × N )) consists exactly of all sequences ( A n ) where A n = P n T ( a ) P n + P n KP n + R n L κ ( n ) R n + G n with a ∈ C ( T ) N × N , K, L i compact, ( G n ) ∈ G , and κ ( n ) is the remainder of n mod N . ( b ) A sequence A = ( A n ) ∈ S ( T ( C )) is stable if and only if W ( A ) := s-lim A n P n and � W i ( A ) := s-lim R nN + i A nN + i R nN + i are invertible. Consequently, S ( T ( C N × N )) is piecewise fractal. 10 / 21
A C ∗ -algebra is called elementary if it is isomorphic to K ( H ) for a Hilbert space H ; dual if it is isomorphic to a direct sum of elementary algebras. 11 / 21
A C ∗ -algebra is called elementary if it is isomorphic to K ( H ) for a Hilbert space H ; dual if it is isomorphic to a direct sum of elementary algebras. Theorem Let A be a unital and piecewise fractal C ∗ -subalgebra of F which contains the ideal G . Then ( A ∩ K ) / G is a dual algebra. 11 / 21
A C ∗ -algebra is called elementary if it is isomorphic to K ( H ) for a Hilbert space H ; dual if it is isomorphic to a direct sum of elementary algebras. Theorem Let A be a unital and piecewise fractal C ∗ -subalgebra of F which contains the ideal G . Then ( A ∩ K ) / G is a dual algebra. Consequences: Lifting theorem, splitting of singular values, formula for α -numbers.... 11 / 21
Example: Continuous functions of Toeplitz operators Let X = [0 , 1] and ( ξ n ) a dense sequence in X . Let S ( X, T ( C )) stand for the smallest closed C ∗ -subalgebra of F which contains all sequences ( P n A ( ξ n ) P n ) where A : X → T ( C ) is a continuous function. Clearly, S ( T ( C )) ⊆ S ( X, T ( C )) . Theorem The algebra S ( X, T ( C )) is quasifractal. 12 / 21
Example: Continuous functions of Toeplitz operators Let X = [0 , 1] and ( ξ n ) a dense sequence in X . Let S ( X, T ( C )) stand for the smallest closed C ∗ -subalgebra of F which contains all sequences ( P n A ( ξ n ) P n ) where A : X → T ( C ) is a continuous function. Clearly, S ( T ( C )) ⊆ S ( X, T ( C )) . Theorem The algebra S ( X, T ( C )) is quasifractal. (Proof: Every subsequence of ( ξ n ) has a convergent subsequence ( ξ η ( n ) ) . The restriction S ( X, T ( C )) η is fractal.) 12 / 21
The fractal variety of an algebra Notation: identify strictly increasing sequences η with their range M = η ( N ) . For a C ∗ -subalgebra A of F , let fr A stand for the set of all infinite subsets M of N such that the restriction A| M is fractal. Call M 1 , M 2 ∈ fr A equivalent if M 1 ∪ M 2 ∈ fr A . Then write M 1 ∼ M 2 . 13 / 21
More recommend
