We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

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...

Given an arbitrary infinite matrix A = {A(i, j)} i,j∈G with entries in {0, 1} and having no identically zero rows, we define an algebra O A as the universal C * -algebra generated by partial isometries subject to conditions that generalize, to the infinite case, those introduced by Cuntz and Krieger for finite matrices. We realize O A as the crossed product algebra for a partial dynamical system and, based on this description, we extend to the infinite case some of the main results known to hold in the finite case, namely the uniqueness theorem, the classification of ideals, and the simplicity criteria. O A is always nuclear and we obtain conditions for it to be unital and purely infinite.

Abstract. Given a partial action α of a group G on an associative algebra A, we consider the crossed product A α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of A α G obtained in the context of C * -algebras. In particular, we prove that A α G is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

We give a new definition for the crossed-product of a C * -algebra A by a *-endomorphism α, which depends not only on the pair (A, α) but also on the choice of a transfer operator. With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz-Krieger algebra O A arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transfer operator.