1,721,249 research outputs found
On the notion of noncommutative submanifold
Full text linkWe review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra A is a quotient algebra B such that all derivations of B can be lifted to A. We will argue that in the case of smooth functions on manifolds every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples
On the pseudo-manifold of quantum states
No full textThere are various statements in the physics literature about the stratification of quantum states, for example into orbits of a unitary group, and about generalized differentiable structures on it. Our aim is to clarify and make precise some of these statements. For A an arbitrary finite-dimensional C*-algebra and U(A) the group of unitary elements of A, we observe that the partition of the state space S(A) into U(A) orbits is not a decomposition and that the decomposition into orbit types is not a stratification (its pieces are not manifolds without boundary), while there is a natural Whitney stratification into matrices of fixed rank. For the latter, when A is a full matrix algebra, we give an explicit description of the pseudo-manifold structure (the conical neighborhood around any point). We then make some comments about the infinite-dimensional case
RETRACTED ARTICLE: The coordinate algebra of a quantum symplectic sphere does not embed into any C*-Algebra
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Twisted Reality and the Second-Order Condition
No full textAn interesting feature of the finite-dimensional real spectral triple (A,H,D,J) of the Standard Model is that it satisfies a “second-order” condition: conjugation by J maps the Clifford algebra ClD(A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence ClD(A) -bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence ClD(A) -bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalence to be implemented by a reality operator J, one has to introduce a “twist” and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples
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