Page 1. Forum Math. 18 (2006), 365–389 DOI 10.1515/FORUM.2006.021 Forum Mathematicum ( de Gruyter... more Page 1. Forum Math. 18 (2006), 365–389 DOI 10.1515/FORUM.2006.021 Forum Mathematicum ( de Gruyter 2006 Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids Pere Ara* and Alberto Facchiniy (Communicated by Rüdiger Göbel) ...
We prove a new extension result for $QB-$rings that allows us to examine extensions of rings wher... more We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques.
Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacenc... more Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\in\Z^{(E_0\times E_0)}$, $n_{ij}=#\{$ arrows from $i$ to $j\}$. Write $N^t_E$ and 1 for the matrices $\in \Z^{(E_0\times E_0\setminus\Sink(E))}$ which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\Z(E)\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra, then there is an exact sequence ($n\in\Z$) \[ K_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow} K_n(R)^{(E_0)}\to K_n(L_R(E))\to K_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also show that for general $R$, the obstruction for having a sequence as above is measured by twisted nil-$K$-groups. If we replace $K$-theory by homotopy algebraic $K$-theory, the obstructions dissapear, and we get, for every ring $R$, a long exact sequence \[ KH_n(R...
In this paper we introduce a new class of $K$-algebras associated with quivers. Given any finite ... more In this paper we introduce a new class of $K$-algebras associated with quivers. Given any finite chain $\mathbf{K}_r: K=K_0\subseteq K_1\subseteq ... \subseteq K_r$ of fields and a chain $\mathbf{E}_r : H_0\subset H_1\subset ... \subset H_r=E^0$ of hereditary saturated subsets of the set of vertices $E^0$ of a quiver $E$, we build the mixed path algebra $P_{\mathbf{K}_r}(E,\mathbf{H}_r)$, the mixed Leavitt path algebra $L_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and the mixed regular path algebra $Q_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and we show that they share many properties with the unmixed species $P_K(E)$, $L_K(E)$ and $Q_K(E)$. Comment: 12 pages
The Realization Problem for (von Neumann) regular rings asks what are the conical refinement mono... more The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M obtained by successive refinements of the identity x_0+y_0=x_0+z_0. This monoid is known to be realizable by the algebra A = K[F] of the monogenic free inverse monoid F, for any choice of field K, but A is not an exchange ring. We show that, for any uncountable field K, M is not realizable by a regular K-algebra, but that a suitable universal localization of A provides...
ABSTRACT We collect some standard definitions, results and examples of the theory of C * -algebra... more ABSTRACT We collect some standard definitions, results and examples of the theory of C * -algebras and von Neumann algebras.
In this article we survey some of the recent goings-on in the classification programme of C$^*$-a... more In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using countably generated Hilbert modules (due to Coward, Elliott and Ivanescu). The passage from one picture to another is presented with full, concise, proofs. We indicate the potential role of the Cuntz semigroup in future classification results, particularly for non-simple algebras.
ABSTRACT Among the classes of bounded linear operators on C*-algebras that allow a fairly detaile... more ABSTRACT Among the classes of bounded linear operators on C*-algebras that allow a fairly detailed analysis are the elementary operators, which are of the form S:x ® åj = 1n aj xbj (x Î A) S:x \mapsto \sum\limits_{j = 1}^n {a_j xb_j (x \in A)} for given n-tuples a = (al,…,an), b = (bl,…,bn) ∈ M(A)n, A a C*algebra. Elementary operators, in the context of matrix algebras, were first studied by Stéphanos and Sylvester around the turn of the 19th century. In the last decades of the 20th century, both spectral and structural properties of elementary operators were thoroughly investigated, and this class of operators has found manifold applications, not only in operator theory, but also in non-commutative algebraic geometry [34] and soliton physics [81], among others. Elementary operators serve as building blocks for more general types of operators, and they comprise both inner derivations and inner automorphisms, which are the topic of the previous chapter.
In this chapter we will investigate how to obtain linear mappings which are compatible with the L... more In this chapter we will investigate how to obtain linear mappings which are compatible with the Lie structure of a c*-algebra from those preserving the associative structure, i.e., Lie derivations from derivations, Lie isomorphisms from isomorphisms. In the case of a boundedly centrally closed C*-algebra, this can simply be achieved through an additive perturbation by a centre- valued trace. Since
ABSTRACT In this chapter, we shall investigate C*-algebras A whose centre is ‘rich’ in comparison... more ABSTRACT In this chapter, we shall investigate C*-algebras A whose centre is ‘rich’ in comparison with the ideal structure of A. If A is a simple unital C*-algebra, the one-dimensional centre Z(A) contains sufficient information. But if A is merely prime, a one-dimensional centre appears to be ‘small’ compared with a possibly large lattice of closed ideals. Yet, in this case, Mloc(A) too is prime, whence dim Z(Mloc(A)) = 1 (Proposition 3.3.2). As it emerges, it is the centre Z(Mloc(A)) of the local multiplier algebra of A rather than Z(A), or Z(M(A)) in the non-unital case, that contains information about properties of operators defined on A which are compatible with ideals of A (such as derivations, automorphisms, elementary operators, etc.). This will be the theme of the subsequent chapters, where it will become evident why the behaviour of these classes of operators on prime C*-algebras is so neat; e.g., th e norm of an inner derivation °a on a prime C*-algebra A simply equals twice the distance from a to Z(M(A)) (Corollary 4.1.21).
Page 1. Forum Math. 18 (2006), 365–389 DOI 10.1515/FORUM.2006.021 Forum Mathematicum ( de Gruyter... more Page 1. Forum Math. 18 (2006), 365–389 DOI 10.1515/FORUM.2006.021 Forum Mathematicum ( de Gruyter 2006 Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids Pere Ara* and Alberto Facchiniy (Communicated by Rüdiger Göbel) ...
We prove a new extension result for $QB-$rings that allows us to examine extensions of rings wher... more We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques.
Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacenc... more Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\in\Z^{(E_0\times E_0)}$, $n_{ij}=#\{$ arrows from $i$ to $j\}$. Write $N^t_E$ and 1 for the matrices $\in \Z^{(E_0\times E_0\setminus\Sink(E))}$ which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\Z(E)\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra, then there is an exact sequence ($n\in\Z$) \[ K_n(R)^{(E_0\setminus\Sink(E))}\stackrel{1-N_E^t}{\longrightarrow} K_n(R)^{(E_0)}\to K_n(L_R(E))\to K_{n-1}(R)^{(E_0\setminus\Sink(E))} \] We also show that for general $R$, the obstruction for having a sequence as above is measured by twisted nil-$K$-groups. If we replace $K$-theory by homotopy algebraic $K$-theory, the obstructions dissapear, and we get, for every ring $R$, a long exact sequence \[ KH_n(R...
In this paper we introduce a new class of $K$-algebras associated with quivers. Given any finite ... more In this paper we introduce a new class of $K$-algebras associated with quivers. Given any finite chain $\mathbf{K}_r: K=K_0\subseteq K_1\subseteq ... \subseteq K_r$ of fields and a chain $\mathbf{E}_r : H_0\subset H_1\subset ... \subset H_r=E^0$ of hereditary saturated subsets of the set of vertices $E^0$ of a quiver $E$, we build the mixed path algebra $P_{\mathbf{K}_r}(E,\mathbf{H}_r)$, the mixed Leavitt path algebra $L_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and the mixed regular path algebra $Q_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and we show that they share many properties with the unmixed species $P_K(E)$, $L_K(E)$ and $Q_K(E)$. Comment: 12 pages
The Realization Problem for (von Neumann) regular rings asks what are the conical refinement mono... more The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M obtained by successive refinements of the identity x_0+y_0=x_0+z_0. This monoid is known to be realizable by the algebra A = K[F] of the monogenic free inverse monoid F, for any choice of field K, but A is not an exchange ring. We show that, for any uncountable field K, M is not realizable by a regular K-algebra, but that a suitable universal localization of A provides...
ABSTRACT We collect some standard definitions, results and examples of the theory of C * -algebra... more ABSTRACT We collect some standard definitions, results and examples of the theory of C * -algebras and von Neumann algebras.
In this article we survey some of the recent goings-on in the classification programme of C$^*$-a... more In this article we survey some of the recent goings-on in the classification programme of C$^*$-algebras, following the interesting link found between the Cuntz semigroup and the classical Elliott invariant and the fact that the Elliott conjecture does not hold at its boldest. We review the construction of this object both by means of positive elements and via its recent interpretation using countably generated Hilbert modules (due to Coward, Elliott and Ivanescu). The passage from one picture to another is presented with full, concise, proofs. We indicate the potential role of the Cuntz semigroup in future classification results, particularly for non-simple algebras.
ABSTRACT Among the classes of bounded linear operators on C*-algebras that allow a fairly detaile... more ABSTRACT Among the classes of bounded linear operators on C*-algebras that allow a fairly detailed analysis are the elementary operators, which are of the form S:x ® åj = 1n aj xbj (x Î A) S:x \mapsto \sum\limits_{j = 1}^n {a_j xb_j (x \in A)} for given n-tuples a = (al,…,an), b = (bl,…,bn) ∈ M(A)n, A a C*algebra. Elementary operators, in the context of matrix algebras, were first studied by Stéphanos and Sylvester around the turn of the 19th century. In the last decades of the 20th century, both spectral and structural properties of elementary operators were thoroughly investigated, and this class of operators has found manifold applications, not only in operator theory, but also in non-commutative algebraic geometry [34] and soliton physics [81], among others. Elementary operators serve as building blocks for more general types of operators, and they comprise both inner derivations and inner automorphisms, which are the topic of the previous chapter.
In this chapter we will investigate how to obtain linear mappings which are compatible with the L... more In this chapter we will investigate how to obtain linear mappings which are compatible with the Lie structure of a c*-algebra from those preserving the associative structure, i.e., Lie derivations from derivations, Lie isomorphisms from isomorphisms. In the case of a boundedly centrally closed C*-algebra, this can simply be achieved through an additive perturbation by a centre- valued trace. Since
ABSTRACT In this chapter, we shall investigate C*-algebras A whose centre is ‘rich’ in comparison... more ABSTRACT In this chapter, we shall investigate C*-algebras A whose centre is ‘rich’ in comparison with the ideal structure of A. If A is a simple unital C*-algebra, the one-dimensional centre Z(A) contains sufficient information. But if A is merely prime, a one-dimensional centre appears to be ‘small’ compared with a possibly large lattice of closed ideals. Yet, in this case, Mloc(A) too is prime, whence dim Z(Mloc(A)) = 1 (Proposition 3.3.2). As it emerges, it is the centre Z(Mloc(A)) of the local multiplier algebra of A rather than Z(A), or Z(M(A)) in the non-unital case, that contains information about properties of operators defined on A which are compatible with ideals of A (such as derivations, automorphisms, elementary operators, etc.). This will be the theme of the subsequent chapters, where it will become evident why the behaviour of these classes of operators on prime C*-algebras is so neat; e.g., th e norm of an inner derivation °a on a prime C*-algebra A simply equals twice the distance from a to Z(M(A)) (Corollary 4.1.21).
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