In this chapter we introduce the idea of Hilbert modules and briefly discuss some useful results on them. For a more detailed account on this subject, the reader is referred to [81], [90], [98] (and [122] for von Neumann modules).

Hilbert C*-modules

A Hilbert space is a complex vector space equipped with a complex-valued inner product. A natural generalization of this is the concept of Hilbert module, which has become quite an important tool of analysis and mathematical physics in recent times.

Definition 4.1.1 Given a *-subalgebra A ⊆ B(h) (where h is a Hilbert space), a semi-Hilbert A-module E is a right A-module equipped with a sesquilinear map 〈., .〉 : E × E → A satisfying 〈x, y〉* = 〈y, x〉, 〈x, ya〉 = 〈x, y〉a and 〈x, x〉 ≥ 0 for x, y ∈ E and a ∈ A. A semi-Hilbert module E is called a pre-Hilbert module if 〈x, x〉 = 0 if and only if x = 0; and it is called a Hilber C*-module if furthermore A is a C*-algebra and E is complete in the norm x → ∥〈x, x〉∥1/2 where ‖.‖ the C*-norm of A.

It is clear that any semi-Hilbert A-module can be made into a Hilbert module in a canonical way: first quotienting it by the ideal {x : 〈x, x〉 = 0} and then completing the quotient.

Let us assume that A is a C*-algebra. The A-valued inner product 〈., .〉 of a Hilbert module shares some of the important properties of usual complexvalued inner product of a Hilbert space, such as the Cauchy–Schwartz inequality, which we prove now.

On the one hand, in almost all the scientific areas, from physical to social sciences, biology to economics, from meteorology to pattern recognition in remote sensing, the theory of classical probability plays a major role and on the other much of our knowledge about the physical world at least is based on the quantum theory [12]. In a way, quantum theory itself is a new kind of theory of probability (in the language of von Neumann and Birkhoff) (see for example [106]) which contains the classical model, and therefore it is natural to extend the other areas of classical probability theory, in particular the theory of Markov processes and stochastic calculus to this quantum model.

There are more than one possible ways (see for example [127]) to construct the above-mentioned extension and in this book we have chosen the one closest to the classical model in spirit, namely that which contains the classical theory as a submodel. This requirement has ruled out any discussion of areas such as free and monotone-probability models. Once we accept this quantum probabilistic model, the ‘grand design’ that engages us is the ‘canonical construction of a *-homomorphic flow (satisfying a suitable differential equation) on a given algebra of observables such that the expectation semigroup is precisely the given contractive semigroup of completely positive maps on the said algebra’.

This problem of ‘dilation’ is here solved completely for the case when the semigroup has a bounded generator, and also for the more general case (of an unbounded generator) with certain additional conditions such as symmetry and/or covariance with respect to a Lie group action.

In this chapter we shall introduce all the basic materials and preliminary notions needed later on in this book.

C* and von Neumann algebras

For the details on the material of this section, the reader may be referred to [125], [40] and [76].

C*-algebras

An abstract normed *-algebra A is said to be a pre C*-algebra if it satisfies the C*-property : ‖x*x‖ = ‖x‖2. If A is furthermore complete under the norm topology, one says that A is a C*-algebra. The famous structure theorem due to Gelfand, Naimark and Segal (GNS) asserts that every abstract C*-algebra can be embedded as a norm-closed *-subalgebra of B(H) (the set of all bounded linear operators on some Hilbert space H). In view of this, we shall fix a complex Hilbert space H and consider a concrete C*-algebra A inside B(H). The algebra A is said to be unital or nonunital depending on whether it has an identity or not. However, even any nonunital C*-algebra always has a net (sequence in case the algebra is separable in the norm topology) of approximate identity, that is, an nondecreasing net eμ of positive elements such that eμa → a for all a ∈ A. Note that the set of compact operators on an infinite dimensional Hilbert space H, to be denoted by K(H), is an example of nonunital C*-algebra.

We now briefly discuss some of the important aspects of C*-algebra theory. First of all, let us mention the following remarkable characterization of commutative C*-algebras.

Watts and Strogatz Model

As explained in more detail in Section 1.3, our next model was inspired by the popular concept of “six degrees of separation,” which is based on the notion that every one in the world is connected to everyone else through a chain of at most six mutual acquaintances. Now an Erdös–Rényi random graph for n = 6 billion people in which each individual has an average of μ = 42.62 friends would have average pairwise distance (log n)/(log μ) = 6, but would have very few triangles, while in social networks if A and B are friends and A and C are friends, then it is fairly likely that B and C are also friends.

To construct a network with small diameter and a positive density of triangles, Watts and Strogatz (1998) started from a ring lattice with n vertices and k edges per vertex, and then rewired each edge with probability p, connecting one end to a vertex chosen at random. This construction interpolates between regularity (p = 0) and disorder (p = 1). The disordered graph is not quite an Erdös–Rényi graph, since the degree of a node is the sum of a Binomial(k, 1/2) and an independent Poisson(k/2).