On C⁎-algebras associated to product systems

Abstract

Let P be a unital subsemigroup of a group G. We propose an approach to C * -algebras associated to product systems over P . We call the C * -algebra of a given product system E its covariance algebra and denote it by A× E P , where A is the coefficient C * -algebra. We prove that our construction does not depend on the embedding P → G and that a representation of A × E P is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz-Nica-Pimsner algebras, Fowler's Cuntz-Pimsner algebra, semigroup C * -algebras of Xin Li and Exel's crossed products by interaction groups.

Tensor algebras of product systems and their C⁎-envelopes

“…Subsequently, the existence of such an algebra was established in many important cases, including (G, P ) = (Z d , N d ), by Carlsen, Larsen, Sims and Vittadello in [11]. In complete generality, the existence of A × X P was verified recently by Sehnem [54]. Sehnem defines the algebra A × X P as a universal object for a particular class of representations of X that she calls strongly covariant and then establishes properties (A) and (B) in [54,Theorem 3.10].…”

“…In complete generality, the existence of A × X P was verified recently by Sehnem [54]. Sehnem defines the algebra A × X P as a universal object for a particular class of representations of X that she calls strongly covariant and then establishes properties (A) and (B) in [54,Theorem 3.10]. This approach has the additional advantage that A × X P admits a more concrete presentation than that of N O r X .…”

“…In what follows the pair (N O r X , j) will be simply denoted as N O r X . An algebra closely related to N O r X is the C * -algebra A × X P of Sehnem [54]. Specialized to compactly aligned product systems, the algebra A × X P is determined by the following properties: there exists a Nica-covariant representation j ′ : X → A × X P that induces a canonical * -homomorphism j ′ * : N T X → A × X P so that (A) j ′ * restricts to an injection on X e = A, and (B) any representation of A× X P which is faithful on j ′ * (A) is also faithful on the fixed point algebra (A × X P ) δ , where δ is the canonical gauge coaction of G on A × X P .…”

“…Let X be a compactly aligned product system over P with coefficients in A. Then the C * -algebras (i) C * e (N T + X ), the C * -envelope of the Nica tensor algebra N T + X , (ii) N O r X , the co-universal C * -algebra for gauge-compatible, Nica-covariant representations of X, of Carlsen, Larsen, Sims and Vittadello [11], and (iii) Sehnem's C * -algebra A × X P [54], are mutually isomorphic via maps that send generators to generators.…”

“…However, it was unclear what the right analogue of the Nica-type Cuntz-Pimsner algebra of a non-injective product system X should be and this was left open for quite some time. Nevertheless this has now been completely settled by Sehnem's covariance algebra A × X P [54]. For a compactly aligned product system X with coefficient algebra A, Sehnem's covariance algebra A × X P plays the role of a Cuntz-Nica-Pimsner algebra, as it is a quotient of N T X containing a faithful copy of X and satisfying a gauge invariance uniqueness theorem.…”