Let A be an AH algebra, that is, A is the inductive limit C^*-algebra of A_1ϕ_1,2A_2ϕ_2,3A_3⟶...⟶... more Let A be an AH algebra, that is, A is the inductive limit C^*-algebra of A_1ϕ_1,2A_2ϕ_2,3A_3⟶...⟶ A_n⟶... with A_n=⊕_i=1^t_nP_n,iM_[n,i](C(X_n,i))P_n,i, where X_n,i are compact metric spaces, t_n and [n,i] are positive integers, and P_n,i∈ M_[n,i](C(X_n,i)) are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that _n,idim(X_n,i)<+∞. In this article, we prove that A can be written as the inductive limit of B_1⟶ B_2⟶...⟶ B_n⟶..., where B_n=⊕_i=1^s_nQ_n,iM_{n,i}(C(Y_n,i))Q_n,i, where Y_n,i are {pt}, [0,1], S^1, T_II, k, T_III, k and S^2 (all of them are connected simplicial complexes of dimension at most three), s_n and {n,i} are positive integers and Q_n,i∈ M_{n,i}(C(Y_n,i)) are projections. This theorem unifies and generalizes the reduction theorem for real rank zero AH algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple...
For an action of a finite group on a C*-algebra, we present some conditions under which propertie... more For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When the group is finite abelian, we prove that crossed products and fixed point algebras preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even for the two element group). The construction also gives an example of a C*-algebra which does not have the ideal property but such that the algebra of 2 by 2 matrices over it does have the ideal property; in fact, th...
We define a Riesz type interpolation property for the Cuntz semigroup of a C^*-algebra and prove ... more We define a Riesz type interpolation property for the Cuntz semigroup of a C^*-algebra and prove it is satisfied by the Cuntz semigroup of every C^*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C^*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite C^*-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C^*-algebra that has (normalized) quasitraces. We prove that large classes of C^*-algebras (including large classes of AH algebras) with the ideal property have this comparison property.
Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*-algebra A. We present some c... more Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*-algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When G is finite abelian, we prove that crossed products and fixed point algebras by G preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even when the group is Z2). The construction also gives an example of a C*-algebra B which does not have the ideal property but such that M2(B) does have the ideal property; in fact, M2(B) has...
For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infini... more For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with real rank zero, and stability under tensoring with a strongly selfabsorbing C*-algebra, we prove the following. Consider an arbitrary action of a second countable compact abelian group on a separable C*-algebra. Then the fixed point algebra under the action has the given property if and only if the crossed product has the same property.
Abstract. A C∗-algebra has the ideal property if any ideal (closed, two-sided) is generated (as a... more Abstract. A C∗-algebra has the ideal property if any ideal (closed, two-sided) is generated (as an ideal) by its projections. We prove a theorem which implies, in particular, that an AH algebra (AH stands for “approximately homogeneous”) stably isomorphic to a C∗-algebra with the ideal property has the ideal property. It is shown that, for any AH algebra A with the ideal property and slow dimension growth, the projections in M∞(A) satisfy the Riesz decomposition and interpolation properties and K0(A) is a Riesz group. We prove a theorem which describes the partially ordered set of all the ideals generated by projections of an AH algebra A; the special case when the projections in M∞(A) satisfy the Riesz decomposition property is also considered. This theorem generalizes a result of G.A. Elliott which gives the ideal structure of an AF algebra. We answer — jointly with M. Dadarlat — a question of G.K. Pedersen, constructing extensions of C∗-algebras with the ideal property which do n...
We give several necessary and sufficient conditions for an AH algebra to have its ideals generate... more We give several necessary and sufficient conditions for an AH algebra to have its ideals generated by their projections. Denote by C the class of AH algebras as above and in addition with slow dimension growth. We completely classify the alge-bras in C up to a shape equivalence by a K-theoretical invari-ant. For this, we show first, in particular, that any C∗-algebra in C is shape equivalent to an AH algebra with slow dimension growth and real rank zero (generalizing so a result of Elliott-Gong); then, we use a classification result of Dadarlat-Gong. We prove that any AH algebra in C has stable rank one (i.e., in the unital case, that the set of the invertible elements is dense in the algebra), generalizing results of Blackadar-Dadarlat-Rørdam and of Elliott-Gong. Other nonstable K-theoretical results for C∗-algebras in C are also proved, generalizing re-sults of Dadarlat-Némethi, Martin-Pasnicu and Blackadar.
Abstract. In this paper we study the C*-algebras associated to continuous fields over locally com... more Abstract. In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants. 1.
Let A be an AH algebra, that is, A is the inductive limit C^*-algebra of A_1ϕ_1,2A_2ϕ_2,3A_3⟶...⟶... more Let A be an AH algebra, that is, A is the inductive limit C^*-algebra of A_1ϕ_1,2A_2ϕ_2,3A_3⟶...⟶ A_n⟶... with A_n=⊕_i=1^t_nP_n,iM_[n,i](C(X_n,i))P_n,i, where X_n,i are compact metric spaces, t_n and [n,i] are positive integers, and P_n,i∈ M_[n,i](C(X_n,i)) are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that _n,idim(X_n,i)<+∞. In this article, we prove that A can be written as the inductive limit of B_1⟶ B_2⟶...⟶ B_n⟶..., where B_n=⊕_i=1^s_nQ_n,iM_{n,i}(C(Y_n,i))Q_n,i, where Y_n,i are {pt}, [0,1], S^1, T_II, k, T_III, k and S^2 (all of them are connected simplicial complexes of dimension at most three), s_n and {n,i} are positive integers and Q_n,i∈ M_{n,i}(C(Y_n,i)) are projections. This theorem unifies and generalizes the reduction theorem for real rank zero AH algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple...
For an action of a finite group on a C*-algebra, we present some conditions under which propertie... more For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When the group is finite abelian, we prove that crossed products and fixed point algebras preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even for the two element group). The construction also gives an example of a C*-algebra which does not have the ideal property but such that the algebra of 2 by 2 matrices over it does have the ideal property; in fact, th...
We define a Riesz type interpolation property for the Cuntz semigroup of a C^*-algebra and prove ... more We define a Riesz type interpolation property for the Cuntz semigroup of a C^*-algebra and prove it is satisfied by the Cuntz semigroup of every C^*-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C^*-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite C^*-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital C^*-algebra that has (normalized) quasitraces. We prove that large classes of C^*-algebras (including large classes of AH algebras) with the ideal property have this comparison property.
Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*-algebra A. We present some c... more Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*-algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When G is finite abelian, we prove that crossed products and fixed point algebras by G preserve topological dimension zero with no condition on the action. We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even when the group is Z2). The construction also gives an example of a C*-algebra B which does not have the ideal property but such that M2(B) does have the ideal property; in fact, M2(B) has...
For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infini... more For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with real rank zero, and stability under tensoring with a strongly selfabsorbing C*-algebra, we prove the following. Consider an arbitrary action of a second countable compact abelian group on a separable C*-algebra. Then the fixed point algebra under the action has the given property if and only if the crossed product has the same property.
Abstract. A C∗-algebra has the ideal property if any ideal (closed, two-sided) is generated (as a... more Abstract. A C∗-algebra has the ideal property if any ideal (closed, two-sided) is generated (as an ideal) by its projections. We prove a theorem which implies, in particular, that an AH algebra (AH stands for “approximately homogeneous”) stably isomorphic to a C∗-algebra with the ideal property has the ideal property. It is shown that, for any AH algebra A with the ideal property and slow dimension growth, the projections in M∞(A) satisfy the Riesz decomposition and interpolation properties and K0(A) is a Riesz group. We prove a theorem which describes the partially ordered set of all the ideals generated by projections of an AH algebra A; the special case when the projections in M∞(A) satisfy the Riesz decomposition property is also considered. This theorem generalizes a result of G.A. Elliott which gives the ideal structure of an AF algebra. We answer — jointly with M. Dadarlat — a question of G.K. Pedersen, constructing extensions of C∗-algebras with the ideal property which do n...
We give several necessary and sufficient conditions for an AH algebra to have its ideals generate... more We give several necessary and sufficient conditions for an AH algebra to have its ideals generated by their projections. Denote by C the class of AH algebras as above and in addition with slow dimension growth. We completely classify the alge-bras in C up to a shape equivalence by a K-theoretical invari-ant. For this, we show first, in particular, that any C∗-algebra in C is shape equivalent to an AH algebra with slow dimension growth and real rank zero (generalizing so a result of Elliott-Gong); then, we use a classification result of Dadarlat-Gong. We prove that any AH algebra in C has stable rank one (i.e., in the unital case, that the set of the invertible elements is dense in the algebra), generalizing results of Blackadar-Dadarlat-Rørdam and of Elliott-Gong. Other nonstable K-theoretical results for C∗-algebras in C are also proved, generalizing re-sults of Dadarlat-Némethi, Martin-Pasnicu and Blackadar.
Abstract. In this paper we study the C*-algebras associated to continuous fields over locally com... more Abstract. In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants. 1.
Uploads
Papers by Cornel Pasnicu