Following some recent results, we consider here a particular class of states on a O^∗-algebra which we call eigenstates of an element A∈, and we deduce some of their properties. Then, we apply our definition to a family of ladder elements, i.e. elements of obeying certain algebraic (commutation) relations which are physically motivated, and we discuss several results for these states
Let . We prove that contains pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of and to have an infinite-dimensional reflexive quotient. This is optimal and answers Problem 24 of Gonz & aacute;lez and Kania [Japanese Journal of Mathematics 16 (2021), 247-313]. We also show that the relative -Grothendieck property depends on the specified copy of and is not an isomorphic invariant of the subspace. More generally, for every closed there is an -Grothendieck subspace isomorphic to ; taking answers Problem 1 of Gonz & aacute;lez et al. [Journal of Mathematical Analysis and Applications 497 (2021), Article 124857].
ABSTRACT We establish rigidity of ‐quasi‐isometric embeddings between spaces of the form , where is a uniformly convex Banach space that is uniformly p‐antipodally bounded . By quasi‐isometric rigidity (or qi‐rigidity ), we mean that any such embedding induces a linear isometry from into for some locally compact Hausdorff space . More precisely, if satisfies for all , with and , then the condition guarantees qi‐rigidity, where is the ‐antipodal constant of and is the associated auxiliary ‐antipodal constant. This theorem was previously known only for the special case where is a Hilbert space. The present work extends this qi‐rigidity result to a broader class, including all spaces for . This extension also points toward open questions regarding the relationship between qi‐rigidity and the general geometry of ‐uniformly convex spaces.
In this paper, we introduce the concept of nonuniform -exponential dichotomies to formulate an asymptotic behavior weaker than the known nonuniform -exponential dichotomy. We discuss the roughness of such dichotomies on the whole line and the half line separately without global invertibility of the evolution family. Our results extend the known results to nonuniform -exponential dichotomies in the case that or .
ABSTRACT We introduce a notion of labeled cycle for normal labeled spaces and prove a reduction theorem for Leavitt path algebras associated with labeled graphs. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labeled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz–Krieger uniqueness theorem for Leavitt path algebras associated with labeled graphs, and show that these algebras are semiprime and semiprimitive over fields.
