Relative Cuntz Pimsner Research Articles

Graded K-theory and K-homology of relative Cuntz–Pimsner algebras and graph C⁎-algebras

We establish exact sequences in KK-theory for graded relative Cuntz–Pimsner algebras associated to nondegenerate C⁎-correspondences. We use this to calculate the graded K-theory and K-homology of relative Cuntz–Krieger algebras of directed graphs for gradings induced by {0,1}–valued labellings of their edge sets.

A bicategorical interpretation for relative Cuntz-Pimsner algebras

We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.

C∗-Algebras for partial product systems over N

We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C∗-algebras and relative Cuntz-Pimsner algebras for them and show that the section C∗-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C∗-algebra.

Topological freeness for C⁎-correspondences

We study conditions that ensure uniqueness theorems of Cuntz-Krieger type for relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ associated to a $C^*$-correspondence $X$ over a $C^*$-algebra $A$. We give general sufficient conditions phrased in terms of a multivalued map $\widehat{X}$ acting on the spectrum $\widehat{A}$ of $A$. When $X(J)$ is of Type I we construct a directed graph dual to $X$ and prove a uniqueness theorem using this graph. When $X(J)$ is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for $\mathcal{O}(J,X)$, as well as to an algebraic condition, which we call $J$-acyclicity of $X$. As an application we improve the Fowler-Raeburn uniqueness theorem for the Toeplitz algebra $\mathcal{T}_X$. We give new simplicity criteria for $\mathcal{O}_X$. We generalize and enhance uniqueness results for relative quiver $C^*$-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms.

Morita equivalence of C*-correspondences passes to the related operator algebras

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz–Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Δ-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an application we show that strong Morita equivalence coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.

A note on the gauge invariant uniqueness theorem for C*-correspondences

We present a short proof of the gauge invariant uniqueness theorem for relative Cuntz–Pimsner algebras of C*-correspondences.

Ordered *-Semigroups and a C<sup>*</sup>-Correspondence for a Partial Isometry

Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and ordered and matricially ordered $*$-semigroups are introduced, along with their universal $C^*$-algebras. The universal $C^*$-algebra generated by a partial isometry is isomorphic to a relative Cuntz-Pimsner $C^*$-algebra of a $C^*$-correspondence over the $C^*$-algebra of a matricially ordered $*$-semigroup. One may view the $C^*$-algebra of a partial isometry as the crossed product algebra associated with a dynamical system defined by a complete order map modelled by a partial isometry acting on a matricially ordered $*$-semigroup.

Crossed products by endomorphisms and reduction of relations in relative Cuntz–Pimsner algebras

Starting from an arbitrary endomorphism α of a unital C⁎-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C⁎-dynamical system (A,α) but also on the choice of an ideal J orthogonal to kerα. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz–Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C⁎-correspondence to the ‘smallest’ C⁎-correspondence yielding the same relative Cuntz–Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to the reduction of C⁎-dynamical systems and allows us to establish a coincidence between relative Cuntz–Pimsner algebras and crossed products introduced.

$C^*$-algebras generalizing both relative Cuntz-Pimsner and Doplicher-Roberts algebras

We introduce and analyse the structure of C*-algebras arising from ideals in right tensor C*-precategories, which naturally generalize both relative Cuntz-Pimsner and Doplicher-Roberts algebras. We establish an explicit intrinsic construction of the considered algebras and a number of key results such as structure theorem, gauge-invariant uniqueness theorem or description of the gauge-invariant ideal structure. In particular, these statements give a new insight into the corresponding results for relative Cuntz-Pimsner algebras and are applied to Doplicher-Roberts associated to C*-correspondences.

Simple Cuntz–Pimsner rings

Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz–Pimsner ring contains a non-zero graded ideal, when a relative Cuntz–Pimsner ring is simple, and when every ideal in a relative Cuntz–Pimsner ring is graded, are given. A “Cuntz–Krieger uniqueness theorem” for relative Cuntz–Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz–Pimsner rings are introduced.As applications of these results, a uniqueness result for the Toeplitz algebra of a directed graph and characterizations of when crossed products of a ring by a single automorphism and fractional skew monoid rings of a single corner isomorphism are simple, are obtained.

Crossed products by abelian semigroups via transfer operators

AbstractWe propose a generalization of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz–Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz–Pimsner algebras of product systems.

Exel's crossed product and relative Cuntz–Pimsner algebras

We consider Exel's new construction of a crossed product of a into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.

Adding tails to $C^*$-correspondences

We describe a method of adding tails to C^* -correspondences which generalizes the process used in the study of graph C^* -algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to C^* -algebras associated to general C^* -correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph C^* -algebras and show that properties of these C^* -algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.

Representations of Cuntz-Pimsner algebras

Let X be a Hilbert bimodule over a C*-algebra A. We analyse the structure of the associated Cuntz-Pimsner algebra O X and related algebras using representation-theoretic methods. In particular, we study the ideals L(I) in O x induced by appropriately invariant ideals I in A, and identify the quotients O X /L(I) as relative Cuntz-Pimsner algebras of Muhly and Solel. We also prove a gauge-invariant uniqueness theorem for O X , and investigate the relationship between O X and an alternative model proposed by Doplicher, Pinzari and Zuccante.