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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∗ = x2 for every x ∈ A. For a locally compact Hausdorff space X, the space C0(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∗ = x2 for every x ∈ A. For a locally compact Hausdorff space X, the space C0(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 C0(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(C0(X)) ∼ = C(βX)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 3 / 21

Theorem (Gelfand) Every commutative C*-algebra is *-isomorphic to C0(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(C0(X)) ∼ = C(βX)).

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 pAp = Cp. 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 IAt(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 pAp = Cp. 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 IAt(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 pAp = Cp. 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 IAt(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 IAt(A) is an ideal of A, 2 IAt(A) is isomorphic to a subalgebra of K(H) of compact operators

• n a Hilbert space H,

3 IAt(A) contains all ideals of A which are isomorphic to a subalgebra

• f K(H).

Definition A C ∗-algebra A is called scattered if for every nonzero C ∗-subalgebra B ⊆ A, the ideal IAt(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 IAt(A) is an ideal of A, 2 IAt(A) is isomorphic to a subalgebra of K(H) of compact operators

• n a Hilbert space H,

3 IAt(A) contains all ideals of A which are isomorphic to a subalgebra

• f K(H).

Definition A C ∗-algebra A is called scattered if for every nonzero C ∗-subalgebra B ⊆ A, the ideal IAt(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 Σn∈Ntnµn where µns

are pure states and tn ∈ R+ ∪ {0} are such that Σn∈Ntn < ∞.

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 (Jα)α≤m(A) such that J0 = {0}, Jm(A) = A and Jα+1/Jα 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 (Iα)α<ht(A) of ideals of A by induction. Put I0 = {0}. At successor stage α + 1: if A/Iα is non-zero, then IAt(A/Iα) is non-zero. Let Iα+1 = σ−1

α (IAt(A/Iα)), where σα : A → A/Iα is the

quotient map. If γ is a limit ordinal let Iα =

α<γ Iα.

The height ht(A) of A, is the smallest ordinal α such that Iα = 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 (Iα)α<ht(A) of ideals of A by induction. Put I0 = {0}. At successor stage α + 1: if A/Iα is non-zero, then IAt(A/Iα) is non-zero. Let Iα+1 = σ−1

α (IAt(A/Iα)), where σα : A → A/Iα is the

quotient map. If γ is a limit ordinal let Iα =

α<γ Iα.

The height ht(A) of A, is the smallest ordinal α such that Iα = 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 (Iα)α<ht(A) of ideals of A by induction. Put I0 = {0}. At successor stage α + 1: if A/Iα is non-zero, then IAt(A/Iα) is non-zero. Let Iα+1 = σ−1

α (IAt(A/Iα)), where σα : A → A/Iα is the

quotient map. If γ is a limit ordinal let Iα =

α<γ Iα.

The height ht(A) of A, is the smallest ordinal α such that Iα = 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 (Iα)α<ht(A) of ideals of A by induction. Put I0 = {0}. At successor stage α + 1: if A/Iα is non-zero, then IAt(A/Iα) is non-zero. Let Iα+1 = σ−1

α (IAt(A/Iα)), where σα : A → A/Iα is the

quotient map. If γ is a limit ordinal let Iα =

α<γ Iα.

The height ht(A) of A, is the smallest ordinal α such that Iα = 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 (Iα)α<ht(A) of ideals of A by induction. Put I0 = {0}. At successor stage α + 1: if A/Iα is non-zero, then IAt(A/Iα) is non-zero. Let Iα+1 = σ−1

α (IAt(A/Iα)), where σα : A → A/Iα is the

quotient map. If γ is a limit ordinal let Iα =

α<γ Iα.

The height ht(A) of A, is the smallest ordinal α such that Iα = 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 (Iα)α≤ht(C(K)) satisfying ht(C(K)) = ht(K), Iα = {f ∈ C(K) : f |K (α) = 0}, IAt(C(K)/Iα) ∼ = c0(K (α) \ K (α+1)).

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 (Iα)α≤ht(C(K)) satisfying ht(C(K)) = ht(K), Iα = {f ∈ C(K) : f |K (α) = 0}, IAt(C(K)/Iα) ∼ = c0(K (α) \ K (α+1)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 8 / 21

Ψ-spaces (Mr´

• wka-Isbell)

Let D = {Aξ : ξ < κ} be an almost disjoint family of subsets of N of size κ. The Ψ(D) is the space N ∪ D, where all elements of N are isolated and the basic neighborhoods of Aξ ∈ D are of the form {Aξ} ∪ Aξ \ F for some finite set F ⊆ N. Ψ(D) is a separable, scattered space of height two. Therefore for C0(Ψ(D)) we have I1 = c0, I2/I1 ∼ = c0(κ), I2 = C0(Ψ(D)). Hence satisfies the short exact sequence 0 → c0

ι

֒ − → C0(Ψ(D)) → c0(κ) → 0, and ι[c0] is an essential ideal of C0(Ψ(D)), i.e., it has non-zero intersection with any non-zero ideal, i.e., N is a dense open subset of C0(Ψ(D)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 9 / 21

Ψ-spaces (Mr´

• wka-Isbell)

Let D = {Aξ : ξ < κ} be an almost disjoint family of subsets of N of size κ. The Ψ(D) is the space N ∪ D, where all elements of N are isolated and the basic neighborhoods of Aξ ∈ D are of the form {Aξ} ∪ Aξ \ F for some finite set F ⊆ N. Ψ(D) is a separable, scattered space of height two. Therefore for C0(Ψ(D)) we have I1 = c0, I2/I1 ∼ = c0(κ), I2 = C0(Ψ(D)). Hence satisfies the short exact sequence 0 → c0

ι

֒ − → C0(Ψ(D)) → c0(κ) → 0, and ι[c0] is an essential ideal of C0(Ψ(D)), i.e., it has non-zero intersection with any non-zero ideal, i.e., N is a dense open subset of C0(Ψ(D)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 9 / 21

Ψ-spaces (Mr´

• wka-Isbell)

Let D = {Aξ : ξ < κ} be an almost disjoint family of subsets of N of size κ. The Ψ(D) is the space N ∪ D, where all elements of N are isolated and the basic neighborhoods of Aξ ∈ D are of the form {Aξ} ∪ Aξ \ F for some finite set F ⊆ N. Ψ(D) is a separable, scattered space of height two. Therefore for C0(Ψ(D)) we have I1 = c0, I2/I1 ∼ = c0(κ), I2 = C0(Ψ(D)). Hence satisfies the short exact sequence 0 → c0

ι

֒ − → C0(Ψ(D)) → c0(κ) → 0, and ι[c0] is an essential ideal of C0(Ψ(D)), i.e., it has non-zero intersection with any non-zero ideal, i.e., N is a dense open subset of C0(Ψ(D)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 9 / 21

Transition from topology to C*-algebras

For a set X, consider the Hilbert space ℓ2(X) of density |X|, with the canonical orthonormal basis {ex : x ∈ X}. Let ℓ2 = ℓ2(N). Faithfully represent C0(Ψ(D)) in B(ℓ2), by π : C0(Ψ(D)) → B(ℓ2) π(χ{n}) = Proj span{en}, π(χAξ) = Proj span{en : n ∈ Aξ}. Let Pξ = π(χAξ). π[C(Ψ(D))] is the commutative C*-algebra generated by {Pξ : ξ < κ} and c0 ⊆ K(ℓ2). For ξ = η, we have PξPη ∈ K(H), since Aξ ∩ Aη is finite.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 10 / 21

Transition from topology to C*-algebras

For a set X, consider the Hilbert space ℓ2(X) of density |X|, with the canonical orthonormal basis {ex : x ∈ X}. Let ℓ2 = ℓ2(N). Faithfully represent C0(Ψ(D)) in B(ℓ2), by π : C0(Ψ(D)) → B(ℓ2) π(χ{n}) = Proj span{en}, π(χAξ) = Proj span{en : n ∈ Aξ}. Let Pξ = π(χAξ). π[C(Ψ(D))] is the commutative C*-algebra generated by {Pξ : ξ < κ} and c0 ⊆ K(ℓ2). For ξ = η, we have PξPη ∈ K(H), since Aξ ∩ Aη is finite.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 10 / 21

Transition from topology to C*-algebras

For a set X, consider the Hilbert space ℓ2(X) of density |X|, with the canonical orthonormal basis {ex : x ∈ X}. Let ℓ2 = ℓ2(N). Faithfully represent C0(Ψ(D)) in B(ℓ2), by π : C0(Ψ(D)) → B(ℓ2) π(χ{n}) = Proj span{en}, π(χAξ) = Proj span{en : n ∈ Aξ}. Let Pξ = π(χAξ). π[C(Ψ(D))] is the commutative C*-algebra generated by {Pξ : ξ < κ} and c0 ⊆ K(ℓ2). For ξ = η, we have PξPη ∈ K(H), since Aξ ∩ Aη is finite.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 10 / 21

Transition from topology to C*-algebras

For a set X, consider the Hilbert space ℓ2(X) of density |X|, with the canonical orthonormal basis {ex : x ∈ X}. Let ℓ2 = ℓ2(N). Faithfully represent C0(Ψ(D)) in B(ℓ2), by π : C0(Ψ(D)) → B(ℓ2) π(χ{n}) = Proj span{en}, π(χAξ) = Proj span{en : n ∈ Aξ}. Let Pξ = π(χAξ). π[C(Ψ(D))] is the commutative C*-algebra generated by {Pξ : ξ < κ} and c0 ⊆ K(ℓ2). For ξ = η, we have PξPη ∈ K(H), since Aξ ∩ Aη is finite.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 10 / 21

Definition (Wofsey) For a Hilbert space H, a family P of noncompact projections of B(H) is called almost orthogonal if the product of any two distinct elements of it is a compact operator. Such a family P is called maximal if for every noncompact projection Q ∈ B(H) the operator PQ is noncompact, for some P ∈ P. Suppose D = {Aξ : ξ < κ} an almost disjoint family of subsets of N, then {Pξ : ξ < κ} is an almost disjoint family of projections in B(ℓ2). Observation.(Wofsey) For a maximal almost disjoint family D = {Aξ : ξ < κ}, the corresponding almost orthogonal family of projections {Pξ : ξ < κ} is not necessarily maximal.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 11 / 21

Definition (Wofsey) For a Hilbert space H, a family P of noncompact projections of B(H) is called almost orthogonal if the product of any two distinct elements of it is a compact operator. Such a family P is called maximal if for every noncompact projection Q ∈ B(H) the operator PQ is noncompact, for some P ∈ P. Suppose D = {Aξ : ξ < κ} an almost disjoint family of subsets of N, then {Pξ : ξ < κ} is an almost disjoint family of projections in B(ℓ2). Observation.(Wofsey) For a maximal almost disjoint family D = {Aξ : ξ < κ}, the corresponding almost orthogonal family of projections {Pξ : ξ < κ} is not necessarily maximal.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 11 / 21

Definition (Wofsey) For a Hilbert space H, a family P of noncompact projections of B(H) is called almost orthogonal if the product of any two distinct elements of it is a compact operator. Such a family P is called maximal if for every noncompact projection Q ∈ B(H) the operator PQ is noncompact, for some P ∈ P. Suppose D = {Aξ : ξ < κ} an almost disjoint family of subsets of N, then {Pξ : ξ < κ} is an almost disjoint family of projections in B(ℓ2). Observation.(Wofsey) For a maximal almost disjoint family D = {Aξ : ξ < κ}, the corresponding almost orthogonal family of projections {Pξ : ξ < κ} is not necessarily maximal.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 11 / 21

Theorem (Mr´

• wka)

There is a maximal almost disjoint family D of size c, such that |β(Ψ(D)) \ Ψ(D)| = 1. Therefore the multiplier algebra of C0(Ψ(D)) is isomorphic to its (minimal) unitization. Recall that the commutative Mr´

• wka C*-algebra C0(Ψ(D)) satisfies

0 → c0

ι

֒ − → C0(Ψ(D)) → c0(c) → 0, where ι[c0] is an essential ideal of C0(Ψ(D)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 12 / 21

Theorem (Mr´

• wka)

There is a maximal almost disjoint family D of size c, such that |β(Ψ(D)) \ Ψ(D)| = 1. Therefore the multiplier algebra of C0(Ψ(D)) is isomorphic to its (minimal) unitization. Recall that the commutative Mr´

• wka C*-algebra C0(Ψ(D)) satisfies

0 → c0

ι

֒ − → C0(Ψ(D)) → c0(c) → 0, where ι[c0] is an essential ideal of C0(Ψ(D)).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 12 / 21

We would like to obtain a non-commutative version of the Mr´

• wka

C*-algebra. A C*-algebra A ⊆ B(ℓ2) which contains K(ℓ2) as an essential ideal and satisfies the following short exact sequence 0 → K(ℓ2)

ι

֒ − → A π − → K(ℓ2(c)) → 0, and the multiplier algebra M(A) of A is isomorphic to its minimal unitization (i.e., M(A)/A ∼ = C). In order to do so, we need to add partial isometries to {Pξ : ξ < c} sending the projection Pξ to Pη for each pair ξ, η < c, i.e., elements Tξ,η

• f B(ℓ2) such that Tξ,ηT ∗

ξ,η = Pξ and T ∗ ξ,ηTξ,η = Pη.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 13 / 21

We would like to obtain a non-commutative version of the Mr´

• wka

C*-algebra. A C*-algebra A ⊆ B(ℓ2) which contains K(ℓ2) as an essential ideal and satisfies the following short exact sequence 0 → K(ℓ2)

ι

֒ − → A π − → K(ℓ2(c)) → 0, and the multiplier algebra M(A) of A is isomorphic to its minimal unitization (i.e., M(A)/A ∼ = C). In order to do so, we need to add partial isometries to {Pξ : ξ < c} sending the projection Pξ to Pη for each pair ξ, η < c, i.e., elements Tξ,η

• f B(ℓ2) such that Tξ,ηT ∗

ξ,η = Pξ and T ∗ ξ,ηTξ,η = Pη.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 13 / 21

Definition Suppose P = {Pξ : ξ < κ} is a family of almost orthogonal projections. We say T = {Tξ,η : ξ, η < κ} ⊆ B(ℓ2) is a system of almost matrix units based on P if and only if for every α, β, ξ, η < κ,

1 Tξ,ξ = Pξ, 2 T ∗

η,ξ − Tξ,η is a compact operator,

3 Tβ,α Tη,ξ − δα,ηTβ,ξ is a compact operator.

We say that T is a maximal system of almost unit projections if it is based

• n a maximal family of almost orthogonal projections.

They exist. If T = {Tξ,η : ξ, η < κ} is a system of almost matrix units, let A(T ) denote the C*-subalgebra of B(ℓ2) generated by {Tξ,η : ξ, η < κ} and the compact operators (K(ℓ2)) in B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 14 / 21

Definition Suppose P = {Pξ : ξ < κ} is a family of almost orthogonal projections. We say T = {Tξ,η : ξ, η < κ} ⊆ B(ℓ2) is a system of almost matrix units based on P if and only if for every α, β, ξ, η < κ,

1 Tξ,ξ = Pξ, 2 T ∗

η,ξ − Tξ,η is a compact operator,

3 Tβ,α Tη,ξ − δα,ηTβ,ξ is a compact operator.

We say that T is a maximal system of almost unit projections if it is based

• n a maximal family of almost orthogonal projections.

They exist. If T = {Tξ,η : ξ, η < κ} is a system of almost matrix units, let A(T ) denote the C*-subalgebra of B(ℓ2) generated by {Tξ,η : ξ, η < κ} and the compact operators (K(ℓ2)) in B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 14 / 21

Definition Suppose P = {Pξ : ξ < κ} is a family of almost orthogonal projections. We say T = {Tξ,η : ξ, η < κ} ⊆ B(ℓ2) is a system of almost matrix units based on P if and only if for every α, β, ξ, η < κ,

1 Tξ,ξ = Pξ, 2 T ∗

η,ξ − Tξ,η is a compact operator,

3 Tβ,α Tη,ξ − δα,ηTβ,ξ is a compact operator.

We say that T is a maximal system of almost unit projections if it is based

• n a maximal family of almost orthogonal projections.

They exist. If T = {Tξ,η : ξ, η < κ} is a system of almost matrix units, let A(T ) denote the C*-subalgebra of B(ℓ2) generated by {Tξ,η : ξ, η < κ} and the compact operators (K(ℓ2)) in B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 14 / 21

Definition Suppose P = {Pξ : ξ < κ} is a family of almost orthogonal projections. We say T = {Tξ,η : ξ, η < κ} ⊆ B(ℓ2) is a system of almost matrix units based on P if and only if for every α, β, ξ, η < κ,

1 Tξ,ξ = Pξ, 2 T ∗

η,ξ − Tξ,η is a compact operator,

3 Tβ,α Tη,ξ − δα,ηTβ,ξ is a compact operator.

We say that T is a maximal system of almost unit projections if it is based

• n a maximal family of almost orthogonal projections.

They exist. If T = {Tξ,η : ξ, η < κ} is a system of almost matrix units, let A(T ) denote the C*-subalgebra of B(ℓ2) generated by {Tξ,η : ξ, η < κ} and the compact operators (K(ℓ2)) in B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 14 / 21

A(T ) is a scattered C ∗-algebra of height 2. I1 = IAt(A(T )) = K(ℓ2), I2/I1 = IAt(A(T )/I1) ∼ = K(ℓ2(κ)). Lemma Suppose that T = {Tξ,η : ξ, η < κ} is a system of almost matrix units in B(ℓ2). Then A(T ) satisfies the short exact sequence 0 → K(ℓ2)

ι

֒ − → A(T ) π − → K(ℓ2(κ)) → 0, where K(ℓ2) is an essential ideal of A(T ).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 15 / 21

A(T ) is a scattered C ∗-algebra of height 2. I1 = IAt(A(T )) = K(ℓ2), I2/I1 = IAt(A(T )/I1) ∼ = K(ℓ2(κ)). Lemma Suppose that T = {Tξ,η : ξ, η < κ} is a system of almost matrix units in B(ℓ2). Then A(T ) satisfies the short exact sequence 0 → K(ℓ2)

ι

֒ − → A(T ) π − → K(ℓ2(κ)) → 0, where K(ℓ2) is an essential ideal of A(T ).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 15 / 21

Theorem ( G., Koszmider) There is a system of almost matrix units S of size c such that A(S) has the property that the multiplier algebra of A(S) is isomorphic to the (minimal) unitization of A(S). Let 2N denote the Cantor space. For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}. Then {Aξ : ξ ∈ 2N} is an almost disjoint family of subsets of 2<N of size continuum.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 16 / 21

Theorem ( G., Koszmider) There is a system of almost matrix units S of size c such that A(S) has the property that the multiplier algebra of A(S) is isomorphic to the (minimal) unitization of A(S). Let 2N denote the Cantor space. For each ξ ∈ 2N we can associate a set Aξ = {s ∈ 2<N : s ⊆ ξ}. Then {Aξ : ξ ∈ 2N} is an almost disjoint family of subsets of 2<N of size continuum.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 16 / 21

Let H denote the separable Hilbert space ℓ2(2<N). Then P2N = {Pξ : ξ ∈ 2N} is a family of almost orthogonal projections in B(H). Let {es : s ∈ 2<N} be the canonical orthonormal basis for H, i.e., es(t) = 1 if t = s and es(t) = 0, otherwise. For every ξ, η ∈ 2N, define a linear bounded operator Tη,ξ : H → H by Tη,ξ(eξ|k) = eη|k for every k ∈ N and Tη,ξ(et) = 0 if t is not equal to ξ|k for any k ∈ N. Then T2N = {Tη,ξ : ξ, η ∈ 2N} is a system of almost matrix units based on P2N.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 17 / 21

Let H denote the separable Hilbert space ℓ2(2<N). Then P2N = {Pξ : ξ ∈ 2N} is a family of almost orthogonal projections in B(H). Let {es : s ∈ 2<N} be the canonical orthonormal basis for H, i.e., es(t) = 1 if t = s and es(t) = 0, otherwise. For every ξ, η ∈ 2N, define a linear bounded operator Tη,ξ : H → H by Tη,ξ(eξ|k) = eη|k for every k ∈ N and Tη,ξ(et) = 0 if t is not equal to ξ|k for any k ∈ N. Then T2N = {Tη,ξ : ξ, η ∈ 2N} is a system of almost matrix units based on P2N.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 17 / 21

Let H denote the separable Hilbert space ℓ2(2<N). Then P2N = {Pξ : ξ ∈ 2N} is a family of almost orthogonal projections in B(H). Let {es : s ∈ 2<N} be the canonical orthonormal basis for H, i.e., es(t) = 1 if t = s and es(t) = 0, otherwise. For every ξ, η ∈ 2N, define a linear bounded operator Tη,ξ : H → H by Tη,ξ(eξ|k) = eη|k for every k ∈ N and Tη,ξ(et) = 0 if t is not equal to ξ|k for any k ∈ N. Then T2N = {Tη,ξ : ξ, η ∈ 2N} is a system of almost matrix units based on P2N.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 17 / 21

Let H denote the separable Hilbert space ℓ2(2<N). Then P2N = {Pξ : ξ ∈ 2N} is a family of almost orthogonal projections in B(H). Let {es : s ∈ 2<N} be the canonical orthonormal basis for H, i.e., es(t) = 1 if t = s and es(t) = 0, otherwise. For every ξ, η ∈ 2N, define a linear bounded operator Tη,ξ : H → H by Tη,ξ(eξ|k) = eη|k for every k ∈ N and Tη,ξ(et) = 0 if t is not equal to ξ|k for any k ∈ N. Then T2N = {Tη,ξ : ξ, η ∈ 2N} is a system of almost matrix units based on P2N.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 17 / 21

Extend T2N to a maximal system of almost matrix units T1 = {Tξ,η : 2N ∪ X}, for some set X with |X| ≤ c and X ∩ 2N = ∅. We carefully pair (ξ, η) with (ρ(ξ), ρ(η)) and get new operators Uξ,η = Tξ,η + Tρ(ξ),ρ(η), and form new maximal systems of almost matrix units U of Uξ,ηs. Applying this kind of pairing multiple times, we can obtain a new system S, where any multiplier of A(S) in B(ℓ2) has to belong to A(S) ⊕ C1B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 18 / 21

Extend T2N to a maximal system of almost matrix units T1 = {Tξ,η : 2N ∪ X}, for some set X with |X| ≤ c and X ∩ 2N = ∅. We carefully pair (ξ, η) with (ρ(ξ), ρ(η)) and get new operators Uξ,η = Tξ,η + Tρ(ξ),ρ(η), and form new maximal systems of almost matrix units U of Uξ,ηs. Applying this kind of pairing multiple times, we can obtain a new system S, where any multiplier of A(S) in B(ℓ2) has to belong to A(S) ⊕ C1B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 18 / 21

Extend T2N to a maximal system of almost matrix units T1 = {Tξ,η : 2N ∪ X}, for some set X with |X| ≤ c and X ∩ 2N = ∅. We carefully pair (ξ, η) with (ρ(ξ), ρ(η)) and get new operators Uξ,η = Tξ,η + Tρ(ξ),ρ(η), and form new maximal systems of almost matrix units U of Uξ,ηs. Applying this kind of pairing multiple times, we can obtain a new system S, where any multiplier of A(S) in B(ℓ2) has to belong to A(S) ⊕ C1B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 18 / 21

Stable C*-algebras and Extensions

Definition A C*-algebra A is called stable, if A ⊗ K(ℓ2) ∼ = A. For any infinite-dimensional Hilbert space H, K(H) is stable, since K(H) ⊗ K(ℓ2) ∼ = K(H ⊗ ℓ2) ∼ = K(H). The Mr´

• wka C*-algebra A(S) is not stable, since

B(ℓ2) ֒ → M(A(S)) ⊗ B(ℓ2) ⊆ M(A(S) ⊗ K(ℓ2)). However M(A(S)) does not contain a copy of B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 19 / 21

Stable C*-algebras and Extensions

Definition A C*-algebra A is called stable, if A ⊗ K(ℓ2) ∼ = A. For any infinite-dimensional Hilbert space H, K(H) is stable, since K(H) ⊗ K(ℓ2) ∼ = K(H ⊗ ℓ2) ∼ = K(H). The Mr´

• wka C*-algebra A(S) is not stable, since

B(ℓ2) ֒ → M(A(S)) ⊗ B(ℓ2) ⊆ M(A(S) ⊗ K(ℓ2)). However M(A(S)) does not contain a copy of B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 19 / 21

Stable C*-algebras and Extensions

Definition A C*-algebra A is called stable, if A ⊗ K(ℓ2) ∼ = A. For any infinite-dimensional Hilbert space H, K(H) is stable, since K(H) ⊗ K(ℓ2) ∼ = K(H ⊗ ℓ2) ∼ = K(H). The Mr´

• wka C*-algebra A(S) is not stable, since

B(ℓ2) ֒ → M(A(S)) ⊗ B(ℓ2) ⊆ M(A(S) ⊗ K(ℓ2)). However M(A(S)) does not contain a copy of B(ℓ2).

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 19 / 21

Let A and B be C*-algebras. An extension of A by B is a short exact sequence 0 → B

ι

− → E

π

− → A → 0. The goal of extension theory is, given A and B, to describe and classify all extensions of A by B up to a suitable notion of equivalence. Also to know what properties of A and B are inherited to E. It is well-known (Brown- Douglas-Fillmore) that for an extension 0 → K(ℓ2) ι − → E

π

− → A → 0 for separable A and E, the C*-algebra E is stable if and only if A is stable. The non-commutative Mr´

• wka C*-algebra A(S) confirms that this

statement is not true for non-separable C*-algebras, since 0 → K(ℓ2) ι − → A(S) π − → K(ℓ2(c)) → 0, and A(S) is non-stable.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 20 / 21

Let A and B be C*-algebras. An extension of A by B is a short exact sequence 0 → B

ι

− → E

π

− → A → 0. The goal of extension theory is, given A and B, to describe and classify all extensions of A by B up to a suitable notion of equivalence. Also to know what properties of A and B are inherited to E. It is well-known (Brown- Douglas-Fillmore) that for an extension 0 → K(ℓ2) ι − → E

π

− → A → 0 for separable A and E, the C*-algebra E is stable if and only if A is stable. The non-commutative Mr´

• wka C*-algebra A(S) confirms that this

statement is not true for non-separable C*-algebras, since 0 → K(ℓ2) ι − → A(S) π − → K(ℓ2(c)) → 0, and A(S) is non-stable.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 20 / 21

Let A and B be C*-algebras. An extension of A by B is a short exact sequence 0 → B

ι

− → E

π

− → A → 0. The goal of extension theory is, given A and B, to describe and classify all extensions of A by B up to a suitable notion of equivalence. Also to know what properties of A and B are inherited to E. It is well-known (Brown- Douglas-Fillmore) that for an extension 0 → K(ℓ2) ι − → E

π

− → A → 0 for separable A and E, the C*-algebra E is stable if and only if A is stable. The non-commutative Mr´

• wka C*-algebra A(S) confirms that this

statement is not true for non-separable C*-algebras, since 0 → K(ℓ2) ι − → A(S) π − → K(ℓ2(c)) → 0, and A(S) is non-stable.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 20 / 21

Let A and B be C*-algebras. An extension of A by B is a short exact sequence 0 → B

ι

− → E

π

− → A → 0. The goal of extension theory is, given A and B, to describe and classify all extensions of A by B up to a suitable notion of equivalence. Also to know what properties of A and B are inherited to E. It is well-known (Brown- Douglas-Fillmore) that for an extension 0 → K(ℓ2) ι − → E

π

− → A → 0 for separable A and E, the C*-algebra E is stable if and only if A is stable. The non-commutative Mr´

• wka C*-algebra A(S) confirms that this

statement is not true for non-separable C*-algebras, since 0 → K(ℓ2) ι − → A(S) π − → K(ℓ2(c)) → 0, and A(S) is non-stable.

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 20 / 21

Thank you

Saeed Ghasemi (IMPAN) Non-commutative C-B derivatives 21st July 2016 21 / 21