Matrix stability for G-algebras or G-graded algebras guarantees Morita invariance, implying Morita invariance of bivariant algebraic K-theory and kk-equivalence between ℓ^X ⋊ G and ℓ^{X/G} for free actions.
and Ellis E.,Isomorphism conjectures with proper coefficients, Journal of Pure and Applied Algebra218(2014), no
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Matrix stability and Morita invariance
Matrix stability for G-algebras or G-graded algebras guarantees Morita invariance, implying Morita invariance of bivariant algebraic K-theory and kk-equivalence between ℓ^X ⋊ G and ℓ^{X/G} for free actions.