A New Look at The Crossed-Product of a C*-algebra by an Endomorphism

We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism \alpha, which depends not only on the pair (A,\alpha) but also on the choice of a transfer operator (defined in the paper). 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.Comment: 14 pages, Plain Te

Tight and cover-to-join representations of semilattices and inverse semigroups

We discuss the relationship between tight and cover-to-join representations of semilattices and inverse semigroups, showing that a slight extension of the former, together with an appropriate selection of co-domains, makes the two notions equivalent. As a consequence, when constructing universal objects based on them, one is allowed to substitute cover-to-join for tight and vice-versa

The Soft Torus II: A Variational Analysis of Commutator Norms

The field of C*-algebras over the interval [0,2] for which the fibers are the Soft Tori is shown to be continuous. This result is applied to show that any pair of non-commuting unitary operators can be perturbed (in a weak sense) in such a way to decrease the commutator norm. Perturbations in norm are also considered and a characterization is given for pairs of unitary operators which are local minimum points for the commutator norm in the finite dimensional case.Comment: 14 pages, plain TeX, UNM-RE-00

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

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 K-theory. 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.Comment: 38 pages, TeX forma

A Note on the Representation Theory of Fell Bundles

We show that every Fell bundle B over a locally compact group G is "proper" in a sense recently introduced by Ng. Combining our results with those of Ng we show that if B satisfies the "approximation property" then it is amenable in the sense that the full and reduced cross-sectional C*-algebras coincide.Comment: Plain TeX, 5 pages, no figure

Approximately Finite C*-Algebras and Partial Automorphisms

We prove that every AF-algebra is isomorphic to a crossed product of a commutative AF-algebra by a partial automorphism. The case of UHF-algebras is treated in detail.Comment: 10 pages, TeX forma

Interactions

Given a C*-algebra B, a closed *-subalgebra A contained in B, and a partial isometry S in B which "interacts" with A in the sense that S*aS = H(a)S*S and SaS* = V(a)SS*, where V and H are positive linear operators on A, we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an "interaction" as being a pair of maps (V,H) satisfying the derived properties. Starting with an abstract interaction (V,H) over a C*-algebra A we construct a C*-algebra B containing A and a partial isometry S whose "interaction" with A follows the above rules. We then discuss the possibility of constructing a "covariance algebra" from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a "generalized correspondence". Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz-Pimsner algebras.Comment: 38 pages, Plain TeX forma

Partial actions of groups and actions of inverse semigroups

Given a group G, we construct, in a canonical way, an inverse semigroup S(G) associated to G. The actions of S(G) are shown to be in one-to-one correspondence with the partial actions of G, both in the case of actions on a set, and that of actions as operators on a Hilbert space. In other words, G and S(G) have the same representation theory. We show that S(G) governs the subsemigroup of all closed linear subspaces of a G-graded C*-algebra, generated by the grading subspaces. In the special case of finite groups, the maximum number of such subspaces is computed. A ``partial'' version of the group C*-algebra of a discrete group is introduced. While the usual group C*-algebra of finite commutative groups forgets everything but the order of the group, we show that the partial group C*-algebra of the two commutative groups of order four, namely Z/4 and Z/2+Z/2, are not isomorphic.Comment: 15 pages, plain TeX, no figure

Amenability for Fell Bundles

Given a Fell bundle \B, over a discrete group $\Gamma$, we construct its reduced cross sectional algebra C^*_r(\B), in analogy with the reduced crossed products defined for C*-dynamical systems. When the reduced and full cross sectional algebras of \B are isomorphic, we say that the bundle is amenable. We then formulate an approximation property which we prove to be a sufficient condition for amenability. A theory of $\Gamma$-graded C*-algebras possessing a conditional expectation is developed, with an eye on the Fell bundle that one naturally associates to the grading. We show, for instance, that all such algebras are isomorphic to C^*_r(\B), when the bundle is amenable. We also study induced ideals in graded C*-algebras and obtain a generalization of results of Stratila and Voiculescu on AF-algebras, and of Nica on quasi-lattice ordered groups. A brief comment is made on the relevance, to our theory, of a certain open problem in the theory of exact C*-algebras. An application is given to the case of an $F_n$-grading of the Cuntz-Krieger algebras $O_A$, recently discovered by Quigg and Raeburn. Specifically, we show that the Cuntz-Krieger bundle satisfies the approximation property, and hence is amenable, for all matrices $A$ with entries in {0,1}, even if $A$ does not satisfy the well known property (I) studied by Cuntz and Krieger in their paper.Comment: Plain TeX, 31 pages, no figure

Inverse Semigroups and Combinatorial C*-Algebras

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 a semigroupoid, 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 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.Comment: 105 pages, 12 point, some pictex figures. Revised version with one section (section 2) added following a suggestion by the refere