Morita equivalence for crossed products by Hilbert $C^\ast $-bimodules

Abstract

We introduce the notion of the crossed product A X Z of a C *algebra A by a Hilbert C * -bimodule X. It is shown that given a C * -algebra B which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B 0 and B 1 ), then B is isomorphic to the crossed product B 0 B 1 Z. We then present our main result, in which we show that the crossed products A X Z and B Y Z are strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X ⊗ A M M ⊗ B Y as A − B Hilbert C * -bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C * -bimodules.

Ordered *-Semigroups and a C^*-Correspondence for a Partial Isometry

“…We already know that m j,0 ≥ 1. If m j,0 = 1 then, since w j is reduced, v j must be (1) and so w j = (−1, 1).…”

“…The final section shows that a relative Cuntz-Pimsner C * -algebra [18] associated with this C * -correspondence is isomorphic to the universal C * -algebra generated by a partial isometry. Note that a general theorem [1] already implies that the C * -algebra generated by a partial isometry is the Cuntz-Pimsner algebra over the C * -subalgebra generated by the image of the larger semigroup A 0 . The ideal used to describe this relative Cuntz-Pimsner algebra is contained in the standard ideal J E , the largest ideal on which the restriction of the left action for the correspondence E is an injection into compact adjointable maps of the Hilbert module.…”

$C^*$-Алгебры, Ассоциированные С Обратимыми Расширениями Логистических Отображений

“…Общее скрещенное произведение, основанное на отношениях (1.1)-(1.3), построено в [12], [13]. Как показано в [14], оно может рассматриваться как скрещенное произведение на гильбертов бимодуль [15] и, следовательно, является одной из фундаментальных моделей относительной алгебры Кунца-Пимзнера [16], C * -алгебр, ассоциированных с C * -соответствиями [17], [18], и алгебрами Доплихера-Робертса [19], см. также [20].…”

Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

“…Let HB(A) be the set of all A-A-Hilbert bimodule isomorphic classes of A-A-Hilbert bimodules defined in Brown, Mingo and Shen [5] and Abadie, Eilers and Exel [1]. For any A-A-Hilbert bimodule X, we denote by [X] the A-A-Hilbert bimodule isomorphic class of X.…”

“…In this section, we shall consider a crossed product of a C*-algebra by a Hilbert C*-bimodule defined in Abadie, Eilers and Exel [1].…”