ring R is called right principally-injective if every R-homomorphism f:aR→R,a∈R,extends to R,or equivalently,if every system of equations xa=b(a,b∈R)is solvable in R.In this paper we show that for any arbitrary gra...ring R is called right principally-injective if every R-homomorphism f:aR→R,a∈R,extends to R,or equivalently,if every system of equations xa=b(a,b∈R)is solvable in R.In this paper we show that for any arbitrary graph E and for a field K,principally-injective conditions for the Leavitt path algebra LK(E)are equivalent to that graph E being acyclic.We also show that the principally-injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.展开更多
If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution o...If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra LK（E） is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK（E） to be ＊-regular （i.e., regular with proper involution）. This characterization of ＊-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph- theoretic properties of E alone. As a corollary, we show that Handelman＇s conjecture （stating that every ,-regular ring is unit-regular） holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.展开更多
Let A be a row-finite k-graph without sources. We investigate the relationship between the complex Kumjian-Pask algebra KPc（A） and graph algebra C＊（A）. We identify situations in which the Kumjian-Pask algebra is ...Let A be a row-finite k-graph without sources. We investigate the relationship between the complex Kumjian-Pask algebra KPc（A） and graph algebra C＊（A）. We identify situations in which the Kumjian-Pask algebra is equal to the graph algebra, and the conditions in which the Kumjian-Pask algebra is finite-dimensional.展开更多
文摘ring R is called right principally-injective if every R-homomorphism f:aR→R,a∈R,extends to R,or equivalently,if every system of equations xa=b(a,b∈R)is solvable in R.In this paper we show that for any arbitrary graph E and for a field K,principally-injective conditions for the Leavitt path algebra LK(E)are equivalent to that graph E being acyclic.We also show that the principally-injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.
基金supported by the Spanish MEC and Fondos FEDER through project MTM2007-60333the Junta de Andalucía and Fondos FEDER,jointly,through projects FQM-336 and FQM-2467
文摘If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra LK（E） is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK（E） to be ＊-regular （i.e., regular with proper involution）. This characterization of ＊-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph- theoretic properties of E alone. As a corollary, we show that Handelman＇s conjecture （stating that every ,-regular ring is unit-regular） holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.
基金Supported by Universitas Pendidikan Indonesia(Indonesia University of Education) Research Grant-Hibah Dosen Peneliti(Grant No.558/UN.40.8/LT/2012)
文摘Let A be a row-finite k-graph without sources. We investigate the relationship between the complex Kumjian-Pask algebra KPc（A） and graph algebra C＊（A）. We identify situations in which the Kumjian-Pask algebra is equal to the graph algebra, and the conditions in which the Kumjian-Pask algebra is finite-dimensional.
