scholarly journals Automorphisms, Diffeomorphisms and Strong Morita Equivalence of Irrational Rotation $C^*$-Algebras

1989 ◽  
Vol 12 (2) ◽  
pp. 415-427
Author(s):  
Kazunori KODAKA
Keyword(s):  
Morita Equivalence ◽  
Irrational Rotation ◽  
C Algebras ◽  
Strong Morita Equivalence

Strong Morita Equivalence for Heisenberg C*-Algebras and the Positive Cones of Their K 0-Groups

1988 ◽  
Vol 40 (04) ◽  
pp. 833-864 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Heisenberg Group ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Crossed Products ◽  
C Algebras ◽  
Projective Representations ◽  
Discrete Heisenberg Group ◽  
Strong Morita Equivalence

In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C*-algebra is said to be of class i, i ∊ {1, 2, 3}, if the range of any normalized trace on its K 0 group has rank i as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras.

scholarly journals Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

2021 ◽  
Vol 127 (2) ◽  
pp. 317-336
Author(s):  
Kazunori Kodaka
Keyword(s):  
Discrete Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Countable Discrete Group ◽  
Relative Commutant ◽  
Strong Morita Equivalence

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Strong Morita equivalence for the Denjoy C*-Algebras

1988 ◽  
Vol 31 (4) ◽  
pp. 439-447
Author(s):  
Ian F. Putnam
Keyword(s):  
Periodic Points ◽  
Complete Classification ◽  
Morita Equivalence ◽  
Constant Function ◽  
Necessary And Sufficient Condition ◽  
C Algebras ◽  
Necessary And Sufficient ◽  
Strong Morita Equivalence ◽  
Function Construction

AbstractThe C*-algebras associated with irrational rotations of the circle were classified up to strong Morita equivalence by M. A. Rieffel. As a corollary, he gave a complete classification of the C*-algebras arising from irrational or Kronecker flows on the 2-torus up to *-isomorphism. Here, we extend the result to the socalled Denjoy homeomorphisms. Specifically, we give a necessary and sufficient condition for the strong Morita equivalence of two C*-algebras arising from homeomorphisms of the circle without periodic points. As a corollary, we show that two C*-algebras arising from flows on the 2-torus obtained from such homeomorphisms by the “flow under constant function” construction are *-isomorphic if and only if the flows themselves are topologically conjugate.

scholarly journals On Morita's fundamental theorem for $C^*$-algebras

2001 ◽  
Vol 88 (1) ◽  
pp. 137 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Closed Subspace ◽  
Morita Equivalence ◽  
Operator Space ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Strong Morita Equivalence

We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Maria Joiţa
Keyword(s):  
Inverse Limit ◽  
London Math ◽  
Morita Equivalence ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Continuous Action ◽  
C Algebras ◽  
Dual Module ◽  
Continuous Inverse ◽  
Strong Morita Equivalence

AbstractWe show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C *-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E *(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E *) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].

scholarly journals Stable isomorphism and strong Morita equivalence ofC∗-algebras

1977 ◽  
Vol 71 (2) ◽  
pp. 349-363 ◽  
Author(s):  
Lawrence Brown ◽  
Philip Green ◽  
Marc Rieffel
Keyword(s):  
Morita Equivalence ◽  
Strong Morita Equivalence
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