Circle Actions on C*-Algebras, Partial Automorphisms, and a Generalized Pimsner-Voiculescu Exact Sequence

Abstract

We introduce a method to study C * -algebras possessing an action of the circle group, from the point of view of its internal structure and its Ktheory. Under relatively mild conditions our structure Theorem shows that any C * -algebra, where an action of the circle is given, arises as the result of a construction that generalizes crossed products by the group of integers.Such a generalized crossed product construction is carried out for any partial automorphism of a C * -algebra, where by a partial automorphism we mean an isomorphism between two ideals of the given algebra.Our second main result is an extension to crossed products by partial automorphisms, of the celebrated Pimsner-Voiculescu exact sequence for K-groups.The representation theory of the algebra arising from our construction is shown to parallel the representation theory for C * -dynamical systems. In particular, we generalize several of the main results relating to regular and covariant representations of crossed products.As in most proofs of the Pimsner-Voiculescu exact sequence and related results ([4], [5], [6], [1]), we derive our exact sequence from the K-theory exact sequence for a suitable Toeplitz extension. The crucial step, as it is often the case, is to show that A has the same K-groups as the Toeplitz algebra. We do so by showing that these algebras are, in fact, KK-equivalent.My first attempt at proving the exactness of the sequence above was, of course, by trying to deduce it from the well known result of Pimsner and Voiculescu. After failing in doing so, I am now tempted to believe that this cannot be done. Our proof is done from scratch and, given a rather involving use of KK-theory, it turns out considerably longer then the available proofs of the original result.We would like to thank Bill Paschke for bringing to our attention his paper [1] with J. Anderson, where they generalize a result, from unpublished lecture material of Arveson's, as well as from Proposition (5.5) in [6], from which the crucial step in [17] follows. Our generalization of these ideas plays a central role in the proof of our result.One intersting aspect, central in our use of KK-theory, is worth mentioning here. If A and B are C * -algebras, then KK(A, B) may be described, as was shown by Cuntz [7], by the set of homotopy classes of homomorphisms from qA to the multiplier algebra of B ⊗ K. Nevertheless the KK-theory elements that we need to introduce have no easy description in such terms. Instead we exhibit these elements by replacing B ⊗ K, above, by an algebra which contains B as a full corner and hence is stably isomorphic to B (at least in the separable case). circle actions and partial automorphismsis commutative. Moreover, if A is separable then j induces a KK-equivalence.Proof. LetJ denote J with an added unit and let y = 1 + ab be an invertible element iñ J with a and b in J. The K 1 -class of y being denoted by [y] 1 , we haveRecall from Lemma (1.1) in [14] that the K 1 classes of 1 + rs and 1 + sr coincide, whenever 1 + rs is invertible. So we hav...

Топологически Свободные Частичные Действия И Точные Представления Скрещенных Произведений

“…Будем говорить, что частичное действие {α g } g∈G тополо-гически свободно, если для каждого конечного множества {g 1 …”

“…Понятие скрещенного произведения C * -алгебры на частичное действие груп-пы Z частичными автоморфизмами введено Экселем [1]. Оно было распростра-нено Мак-Кланаханом [2] на случай частичных действий дискретных групп.…”

𝐶*-algebras associated with branched coverings

“…These examples include some of the algebras considered by R. Exel in [9], in the case of a partial homeomorphism.…”

“…g) If σ : U → V is a homeomorphism between two open subsets of X, then we are in the situation considered by Exel in [9], by taking…”

Twisted partial actions of Hopf algebras

“…The desire to endow important classes of C * -algebras generated by partial isometries with a structure of a more general crossed product led to the concept of a partial group action, introduced in [19], [27], [22], [23]. The new structure permitted to obtain relevant results on K-theory, ideal structure and representations of the algebras under consideration, as well as to treat amenability questions, especially amenability of C * -algebraic bundles (also called Fell bundles), using both partial actions and the related concept of a partial representation.…”