Interacting Fock Spaces - Revisited

Interacting Fock spaces are the most general N0–graded (pre-)Hilbert spaces with creation operators that have degree 1 and generate everything out of a single vacuum vector. It is the creators alone that generate the space out of the vacuum; so the same is true for the nonselfadjoint operator algebra generated by the creators. A formal definition has been given by Accardi, Lu, and Volovich (1997). Forthcoming work by Accardi and Skeide (2008), gave a different but equivalent definition, and also several desirable properties (embeddability, and what we are going to call here regularity, but also embeddability in Cuntz-Pimsner-Toeplitz type algebras) have been pointed out there. We will present our latest results on interacting Fock spaces with Malte Gerhold. Every interacting Fock space is embeddable - provided we ask the question the right way. The same is not true for regularity: There are non-regular interacting Fock spaces - non-regular beyond repair; and they are quite natural. We answer the long standing question, when a regular interacting Fock space has bounded generators (giving, thus, rise to embeddings into tensor algebras and Cuntz-Pimsner-Toeplitz algebras). We show how the operator algebras associated with subproduct systems, introduced by Shalit and Solel, and studied intensely by Shalit et al. (with Davidson and Ramsey, and with Kakariadis) fit into the framework of interacting Fock spaces. This raises several questions and proposals for generalizations.