The maximal C*-algebra of quotients as an operator bimodule

We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra A as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product of A with itself labelled by the essential closed right ideals of A into A. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.

Maximal C*-algebras of quotients and injective envelopes of C*-algebras

A new C*-enlargement of a C*-algebra A nested between the local multiplier algebra of A and its injective envelope is introduced. Various aspects of this maximal C*-algebra of quotients are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra such that its second iterated local multiplier algebra is strictly larger than its local multiplier algebra.

Quotients, exactness, and nuclearity in the operator system category

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

Characterisations of the weak expectation property

We use representations of operator systems as quotients to deduce various characterisations of the weak expectation property (WEP) for C∗ -algebras. By Kirchberg’s work on WEP, these results give new formulations of Connes’ embedding problem.

Quantum Chromatic Numbers via Operator Systems

We define several new types of quantum chromatic numbers of a graph and characterize them in terms of operator system tensor products. We establish inequalities between these chromatic numbers and other parameters of graphs studied in the literature and exhibit a link between them and non-signalling correlation boxes.