Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ -modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pureinjective module if k is a countable field.
Let k be a field of characteristic different from 2. We consider two important tame non-polynomial growth algebras: the incidence k-algebra of the garland G3 of length 3 and the incidence k-algebra of the enlargement of the Nazarova-Zavadskij poset N Z by a greatest element. We show that if Λ is one of these algebras, then there exists a special family of pointed Λ-modules, called an independent pair of dense chains of pointed modules. Hence, by a result of Ziegler, Λ admits a super-decomposable pureinjective module if k is a countable field.
Maps of the form Phi(X) =sum_{i=1}^s A_iXA^*, where A_1, . . . ,A_s are fixed complex n by n matrices and X is any complex n by n matrix are used in quantum information theory as representations of quantum channels. This article deals with computable conditions for the existence of decoherence--free subspaces for Phi. Since the definition of decoherence-free subspace for quantum channels relies only on the matrices A1, . . . ,As, the term of common reducing unitary subspace is used instead of the original one. Among the main results of the paper, there are computable conditions for the existence of common eigenvectors. These are related to common reducing unitary subspaces of dimension one. The new results on common eigenvectors provide new effective condition for the existence of common invariant subspaces of arbitrary dimensions.
Assume that A 1 , ..., A s are complex n×n matrices. We give a computable criterion for existence of a common eigenvector of A i which generalize the result of D. Shemesh established for two matrices. We use this criterion to prove some necessary and sufficient condition for A i to have a common invariant subspace of dimension d, 2 ≤ d < n, if every A i has pairwise different eigenvalues. Finally, we observe that the set of all matrices having multiple eigevalues has Lebesgue measure 0 and thus the condition is sufficient in practical applications. Being motivated by quantum information theory, we give a flavour of such applications for irreducible completely positive superoperators.
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