The Structure of C*-Algebras of Product Systems

A major trend in the area of operator theory is the use of operator algebras for quantizing geometrical structures and translating discrete properties of the structures into analytic properties of the related operator algebras. The motivation is two-fold. On the one hand, it can be used to test conjectures within the field: finding operator algebras with certain properties reduces to producing simpler dynamical systems. On the other hand, interdisciplinary links can emerge. For example, classification problems can be tested through isomorphisms of algebras. To this end, the quantizations must reflect rigidly the geometrical behaviour of the structure.

In our study we consider a quantization via a Fock space construction, similar to what is done in quantum mechanics. In the past 20 years, the community has extensively studied C*-algebras that relate to dynamics evolving in one discrete direction. The obtained class contains previously known constructions arising from graphs and dynamical systems that play a prominent role in the theory of C*-algebras.

Katsura's work has been very influential, allowing for further developments. Let us give a short description of several results to emphasize on the impact of the one-variable Gauge Invariant Uniqueness Theorem (GIUT).

i. Katsura achieved necessary and sufficient conditions for nuclearity of Cuntz-Pimsner algebras. Through a careful analysis, he also produced a 6-term exact sequence for computing its K-theory. The link is provided exactly through the solutions of ideals that define the covariant representations.

ii. The purpose of the tail-adding technique is to dilate a non-injective C*-correspondence to an injective one so that their Cuntz-Pimsner algebras are Morita equivalent. This is in analogy to removing sinks of graphs by adding infinite tails. For C*-correspondences, it was first established by Muhly-Tomforde and later extended by Kakariadis-Katsoulis. With some care, the tail can be chosen to preserve sub-classes.

iii. Morita theory effectively matches representations, and a C*-alternative has been introduced by Rieffel. A similar theory has been developed for non-selfadjoint algebras by Blecher-Muhly-Paulsen and Eleftherakis. Muhly-Solel imported it to the context of C*-correspondences. By using the GIUT it was shown by Muhly-Solel and Eleftherakis-Kakariadis-Katsoulis that Morita equivalence is inherited by the related algebras.

iv. Taking motivation from symbolic dynamics, Muhly-Pask-Tomforde lift the strong shift equivalence to regular C*-correspondences. By using the GIUT on the bipartite inflation, they showed that strong shift equivalence implies Morita equivalence for the Cuntz-Pimsner algebras, but not for the Toeplitz-Pimsner algebras and tensor algebras. This theory has been extended to shift equivalence by Kakariadis-Katsoulis. A key point is the minimal extension of an injective C*-correspondence to an essential bimodule. This was established through a direct limit process similar to injective C*-dynamics.

v. Being a co-universal object, the C*-envelope often coincides with a Cuntz-type algebra. This is the case for the tensor algebras, as shown by Katsoulis-Kribs, but this is not exclusive. As shown by Kakariadis-Shalit, the C*-envelope of the tensor algebra of a factorial language is the quotient by generalized compacts. The use of the GIUT is crucial to achieve these results.

The understanding of the covariant relations for more exotic dynamics had remained a mystery for several years. However, recent developments have allowed us to unlock the correct covariant relations that produce rigid boundary quotients. Our motivation in this project is to explore the multi-variable analogues of the Gauge Invariant Uniqueness Theorem and its vast applications. Our research will be carried at the general level but also at the level of product systems of finite rank.