# Projective Modules in Classical and Quantum Functional Analysis

Homological theory of the “algebras in analysis” exists in at least three different versions. First of all, there is the homological theory of Banach and more general locally convex algebras. This is about 40 years old. However, in the last decade of the previous century, a “homological section” appeared in a new branch of analysis, the so-called quantized functional analysis or, more prosaically, the theory of operator spaces. One of principal features of this theory, as is now widely realized, is the existence of different approaches to the proper quantum version of a bounded bilinear operator. In fact, two such versions are now thought to be most important; each of them has its own relevant tensor product with an appropriate universal property. Accordingly, there are two principal versions of quantized algebras and quantized modules, and this leads to two principal versions of quantized homology.

Thus we have now, in the first decade of the 21st century, three species of topological homology: the traditional (or “classical”) one, and two “quantized” ones.

In these lectures, we shall restrict ourselves by studying, in the framework of these three theories, the fundamental concept of a projective module. This concept is “primus inter pares” among the three recognized pillars of the science of homology: projectivity, injectivity, and flatness. It is this notion that is the cornerstone for every sufficiently developed homological theory, let it be in algebra, topology, or, as in these notes, in functional analysis.

Our initial definitions of projectivity do not go far away from their prototypes in abstract algebra. However, the principal results concern essentially functional-analytic objects. As we shall see, they have, as a rule, no purely algebraic analogues. Moreover,

some phenomena are strikingly different from what algebraists could expect, based on their experience.

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