Hermitian Star Products are Completely Positive Deformations

Bursztyn, Henrique; Waldmann, Stefan

doi:10.1007/s11005-005-4844-3

Cited by 17 publications

(25 citation statements)

References 11 publications

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“…In the spirit of complete positivity, we call a deformation Ꮽ completely positive if, for all n ∈ ‫,ގ‬ the * -algebras M n (Ꮽ) are positive deformations of M n (Ꮽ). We remark that not all hermitian deformations are positive [Bursztyn and Waldmann 2004b].…”

Section: Hermitian Deformation Quantizationmentioning

confidence: 89%

See 1 more Smart Citation

Completely positive inner products and strong Morita equivalence

Bursztyn¹,

Waldmann²

2005

Pacific J. Math.

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122

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We develop a general framework for the study of strong Morita equivalence in which C * -algebras and hermitian star products on Poisson manifolds are treated in equal footing. We compare strong and ring-theoretic Morita equivalences in terms of their Picard groupoids for a certain class of unital * -algebras encompassing both examples. Within this class, we show that both notions of Morita equivalence induce the same equivalence relation but generally define different Picard groups. For star products, this difference is expressed geometrically in cohomological terms.

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Section: Hermitian Deformation Quantizationmentioning

confidence: 89%

“…The proof consists of showing that any hermitian star product can be realized as a subalgebra of a formal Weyl algebra, and then use the results in the symplectic case [Bursztyn and Waldmann 2000b], see [Bursztyn and Waldmann 2004b].…”

Section: B Rigidity Of Properties (I) and (Ii)mentioning

confidence: 99%

Completely positive inner products and strong Morita equivalence

Bursztyn¹,

Waldmann²

2005

Pacific J. Math.

Self Cite

122

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“…Also the global formality map of Dolgushev has this property [18]. Finally, one can show that a Hermitian star product is always a completely positive deformation [15].…”

Section: 24]mentioning

confidence: 90%

“…The way out is to allow only those Hermitian deformations which are completely positive deformations, see e.g. [15] for examples.…”

Section: Morita Equivalencementioning

confidence: 99%

Morita theory in deformation quantization

Waldmann

2011

Bull Braz Math Soc, New Series

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Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, * -Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.

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“…In other words, any Hermitian star product is a positive deformation in the sense of [6]. A proof of this theorem can be found in [11].…”

Section: Rieffel Induction In Deformation Quantizationmentioning

confidence: 93%