TY - JOUR
T1 - Morita invariance of unbounded bivariant K-theory
AU - Kaad, Jens
PY - 2024/10
Y1 - 2024/10
N2 - We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator ∗-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator ∗-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov’s bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C1-versions of well-known C∗-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.
AB - We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator ∗-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator ∗-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov’s bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C1-versions of well-known C∗-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.
KW - Morita equivalence
KW - Operator ∗-algebra
KW - Operator ∗-correspondence
KW - Unbounded bivariant K-theory
KW - Unbounded Kasparov module
KW - Unbounded Kasparov product
U2 - 10.1007/s43034-024-00392-3
DO - 10.1007/s43034-024-00392-3
M3 - Journal article
AN - SCOPUS:85205909525
SN - 2639-7390
VL - 15
JO - Annals of Functional Analysis
JF - Annals of Functional Analysis
IS - 4
M1 - 88
ER -