Quantum versions of vector duality and the exponential law in the framework of the non-matrix approach

It is shown what form is taken by certain fundamental constructions and results of quantum functional analysis (that is, the theory of operator spaces) in the framework of the approach that uses vectors with operator coefficients instead of matrices; that is, one deals with 'non-matrix' quantization of spaces that are in a relation of vector duality. After describing the basic construction we establish, in the framework of the 'non-matrix' approach, the quantum version of the exponential law and the quantum version of the dual associativity law that connects operator functors with tensor product functors. At the end of the paper we consider, as an important concrete example, the 'non-matrix' quantization of an operator algebra on a Hilbert space and the basic properties of this quantization.