Non-commutative Cantor-Bendixson derivatives and scattered - PowerPoint PPT Presentation

Non-commutative Cantor-Bendixson derivatives and scattered C*-algebras Saeed Ghasemi (Joint work with Piotr Koszmider) Institute of Mathematics, Polish Academy of Sciences Transfinite methods in Banach spaces and algebras of operators B¸ edlewo 21st July 2016 Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 1 / 21

Definition A C*-algebra A is a Banach *-algebra on the field of complex numbers which satisfies the C*-identity, i.e., � xx ∗ � = � x � 2 for every x ∈ A . For a locally compact Hausdorff space X , the space C 0 ( X ) with f . g ( x ) = f ( x ) g ( x ) , f ∗ ( x ) := f ( x ) , � f � = sup { f ( x ) : x ∈ X } , is a commutative C*-algebra. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 2 / 21

Definition A C*-algebra A is a Banach *-algebra on the field of complex numbers which satisfies the C*-identity, i.e., � xx ∗ � = � x � 2 for every x ∈ A . For a locally compact Hausdorff space X , the space C 0 ( X ) with f . g ( x ) = f ( x ) g ( x ) , f ∗ ( x ) := f ( x ) , � f � = sup { f ( x ) : x ∈ X } , is a commutative C*-algebra. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 2 / 21

Theorem (Gelfand) Every commutative C*-algebra is *-isomorphic to C 0 ( X ) , for a locally compact Hausdorff space X. B ( H ) - The C*-algebra of all bounded linear operators on a Hilbert space H , K ( H ) - The ideal of all compact operators on H , M ( A ) - The multiplier algebra of the C*-algebra A (e.g., M ( C 0 ( X )) ∼ = C ( β X )). Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 3 / 21

Theorem (Gelfand) Every commutative C*-algebra is *-isomorphic to C 0 ( X ) , for a locally compact Hausdorff space X. B ( H ) - The C*-algebra of all bounded linear operators on a Hilbert space H , K ( H ) - The ideal of all compact operators on H , M ( A ) - The multiplier algebra of the C*-algebra A (e.g., M ( C 0 ( X )) ∼ = C ( β X )). Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 3 / 21

Minimal Projections Definition A projection p in a C ∗ -algebra A is called minimal if p A p = C p . For a commutative C*-algebra C ( X ), minimal projections correspond to the characteristic functions of isolated points of X . For a C*-algebra A , let I At ( A ) denote the ∗ -subalgebra of A generated by its minimal projections. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 4 / 21

Minimal Projections Definition A projection p in a C ∗ -algebra A is called minimal if p A p = C p . For a commutative C*-algebra C ( X ), minimal projections correspond to the characteristic functions of isolated points of X . For a C*-algebra A , let I At ( A ) denote the ∗ -subalgebra of A generated by its minimal projections. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 4 / 21

Minimal Projections Definition A projection p in a C ∗ -algebra A is called minimal if p A p = C p . For a commutative C*-algebra C ( X ), minimal projections correspond to the characteristic functions of isolated points of X . For a C*-algebra A , let I At ( A ) denote the ∗ -subalgebra of A generated by its minimal projections. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 4 / 21

Theorem Suppose that A is a C ∗ -algebra. 1 I At ( A ) is an ideal of A , 2 I At ( A ) is isomorphic to a subalgebra of K ( H ) of compact operators on a Hilbert space H , 3 I At ( A ) contains all ideals of A which are isomorphic to a subalgebra of K ( H ) . Definition A C ∗ -algebra A is called scattered if for every nonzero C ∗ -subalgebra B ⊆ A , the ideal I At ( B ) is nonzero. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 5 / 21

Theorem Suppose that A is a C ∗ -algebra. 1 I At ( A ) is an ideal of A , 2 I At ( A ) is isomorphic to a subalgebra of K ( H ) of compact operators on a Hilbert space H , 3 I At ( A ) contains all ideals of A which are isomorphic to a subalgebra of K ( H ) . Definition A C ∗ -algebra A is called scattered if for every nonzero C ∗ -subalgebra B ⊆ A , the ideal I At ( B ) is nonzero. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 5 / 21

Theorem (Huruya, Jensen, Kusuda, Wojtaszczyk) For a C ∗ -algebra A , the following conditions are equivalent: 1 A is scattered. 2 Every positive functional µ on A is of the form Σ n ∈ N t n µ n where µ n s are pure states and t n ∈ R + ∪ { 0 } are such that Σ n ∈ N t n < ∞ . 3 Every non-zero ∗ -homomorphic image of A has a minimal projection, 4 There is an ordinal m ( A ) and a continuous increasing sequence of ideals ( J α ) α ≤ m ( A ) such that J 0 = { 0 } , J m ( A ) = A and J α +1 / J α is an elementary C ∗ -algebra (i.e., ∗ -isomorphic to the algebra of all compact operators on a Hilbert space) for every α < m ( A ) . 5 The dual spaces C ∗ of separable subalgebras C ⊆ A are separable. 6 A does not contain a copy of the C ∗ -algebra C ([0 , 1]) . 7 The spectrum of every self-adjoint element is countable. Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 6 / 21

The Cantor-Bendixson composition series Suppose A is a scattered C*-algebra. We define the Cantor-Benndixson sequence ( I α ) α< ht ( A ) of ideals of A by induction. Put I 0 = { 0 } . At successor stage α + 1: if A / I α is non-zero, then I At ( A / I α ) is non-zero. Let I α +1 = σ − 1 α ( I At ( A / I α )), where σ α : A → A / I α is the quotient map. If γ is a limit ordinal let I α = � α<γ I α . The height ht ( A ) of A , is the smallest ordinal α such that I α = A . Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 7 / 21

The Cantor-Bendixson composition series Suppose A is a scattered C*-algebra. We define the Cantor-Benndixson sequence ( I α ) α< ht ( A ) of ideals of A by induction. Put I 0 = { 0 } . At successor stage α + 1: if A / I α is non-zero, then I At ( A / I α ) is non-zero. Let I α +1 = σ − 1 α ( I At ( A / I α )), where σ α : A → A / I α is the quotient map. If γ is a limit ordinal let I α = � α<γ I α . The height ht ( A ) of A , is the smallest ordinal α such that I α = A . Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 7 / 21

The Cantor-Bendixson composition series Suppose A is a scattered C*-algebra. We define the Cantor-Benndixson sequence ( I α ) α< ht ( A ) of ideals of A by induction. Put I 0 = { 0 } . At successor stage α + 1: if A / I α is non-zero, then I At ( A / I α ) is non-zero. Let I α +1 = σ − 1 α ( I At ( A / I α )), where σ α : A → A / I α is the quotient map. If γ is a limit ordinal let I α = � α<γ I α . The height ht ( A ) of A , is the smallest ordinal α such that I α = A . Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 7 / 21

The Cantor-Bendixson composition series Suppose A is a scattered C*-algebra. We define the Cantor-Benndixson sequence ( I α ) α< ht ( A ) of ideals of A by induction. Put I 0 = { 0 } . At successor stage α + 1: if A / I α is non-zero, then I At ( A / I α ) is non-zero. Let I α +1 = σ − 1 α ( I At ( A / I α )), where σ α : A → A / I α is the quotient map. If γ is a limit ordinal let I α = � α<γ I α . The height ht ( A ) of A , is the smallest ordinal α such that I α = A . Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 7 / 21

The Cantor-Bendixson composition series Suppose A is a scattered C*-algebra. We define the Cantor-Benndixson sequence ( I α ) α< ht ( A ) of ideals of A by induction. Put I 0 = { 0 } . At successor stage α + 1: if A / I α is non-zero, then I At ( A / I α ) is non-zero. Let I α +1 = σ − 1 α ( I At ( A / I α )), where σ α : A → A / I α is the quotient map. If γ is a limit ordinal let I α = � α<γ I α . The height ht ( A ) of A , is the smallest ordinal α such that I α = A . Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 7 / 21

Suppose that K is a scattered compact Hausdorff space with the Cantor-Bendixson sequence ( K ( α ) ) α ≤ ht ( K ) . Then C ( K ) is a commutative scattered C ∗ -algebra with the Cantor-Bendixson sequence ( I α ) α ≤ ht ( C ( K )) satisfying ht ( C ( K )) = ht ( K ) , I α = { f ∈ C ( K ) : f | K ( α ) = 0 } , = c 0 ( K ( α ) \ K ( α +1) ) . I At ( C ( K ) / I α ) ∼ Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 8 / 21

Suppose that K is a scattered compact Hausdorff space with the Cantor-Bendixson sequence ( K ( α ) ) α ≤ ht ( K ) . Then C ( K ) is a commutative scattered C ∗ -algebra with the Cantor-Bendixson sequence ( I α ) α ≤ ht ( C ( K )) satisfying ht ( C ( K )) = ht ( K ) , I α = { f ∈ C ( K ) : f | K ( α ) = 0 } , = c 0 ( K ( α ) \ K ( α +1) ) . I At ( C ( K ) / I α ) ∼ Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 8 / 21