The Gelfand-Naimark-Segal Construction — Part 1

In this series of blog posts I’ll work through the proof that every C^*-algebra on a may be isomorphically embedded into \mathcal{B}(\mathcal{H}), the C^*-algebra of bounded linear operators on some Hilbert space \mathcal H. This post is meant to give definitions of the objects that we will be studying and possibly some motivation for why this is an especially interesting theorem.

This series of posts deal with operators on Hilbert spaces. We say that \mathcal H is a Hilbert space if it is a vector space (over \mathbb C) equipped with an inner product \langle \cdot, \cdot \rangle: \mathcal H \times \mathcal H \rightarrow \mathbb C (which gives rise to an associated norm || x|| = \sqrt{\langle x, x \rangle}) that is complete with respect to this norm.

Given a linear function between Hilbert spaces \tau: \mathcal H_1 \rightarrow \mathcal H_2, one may define the norm of the operator as the supremum of the ratio of ||\tau x|| and ||x|| as x ranges over all x \in \mathcal H_1, namely

||\tau|| = \sup_{x \neq 0} \frac{||\tau x||}{||x||} (1)

If || \tau||< \infty, then \tau is called bounded. The set of all bounded linear operators between \mathcal H_1 and \mathcal H_2 will be denoted by \mathcal B(\mathcal H_1, \mathcal H_2), or as \mathcal B(\mathcal H_1) if \mathcal H_1 = \mathcal H_2. By multiplying the right hand side of equation (1) by \frac{1/||x||}{1/||x||} one can calculate the operator norm by ||\tau|| = \sup_{||y||=1} ||\tau y||.

A useful theorem connecting bounded linear operators and the inner product on a Hilbert space:

Theorem 1: Given a Hilbert space \mathcal H and some bounded linear operator \tau \in \mathcal B(\mathcal H), there exists a unique operator \tau^* \in \mathcal B(\mathcal H) such that \langle \tau x, y\rangle = \langle x, \tau^* y \rangle. (This operator is called the Hilbert space adjoint of \tau.)

Proof: Suppose that \tau is as above, and choose a fixed but arbitrary x_0 \in \mathcal H. Then one can define a bounded linear functional

f_{x_0} (x) = \langle \tau x, x_0\rangle.

Due to the Riesz Representation theorem there exists some y_1 \in \mathcal H such that f(x) = \langle x, y_0\rangle for some unique y_0 \in \mathcal H. Then we can define $\tau^*$ which associates with each $x_0$ this unique $y_0$; this is linear and it is bounded, so then,

\langle \tau x, y \rangle = \langle x, \tau^* y\rangle.

The presence of this operator adjoint for any bounded linear operator on a Hilbert space makes \mathcal B(\mathcal H) a prototypical example of a C^*-algebra.

Definition: A Banach algebra A is an associative algebra which is also a normed vector space and complete with respect to the norm and ||xy|| \leq ||x||\cdot ||y||. If furthermore there is a map ^*: A \rightarrow A which is an involution, satisfies (\lambda x)^* = \bar \lambda x^*, and has the property that ||a^*a|| = ||a||^2, then A is called a \mathbf{C^*}-algebra.

Looking at this definition, one may notice that, despite us proving the existence of the adjoint operator, whose existence arises as a fundamental property of the the set of bounded operators on a Hilbert space, there is no reference to any Hilbert space nor any invocation of any of the orthogonality properties of the inner product. C^*-algebras were originally investigated in the context of bounded operators on a Hilbert space; however, due to the work of Gelfand and Naimark during the middle of the last century and the Gelfand-Naimark-Siegal Theorem one can consider algebras within the purely abstract definition of C^*-algebras and be assured that there are examples of the algebras in question contained within the very concrete space \mathcal B(\mathcal H).

It is here that we will begin our story in the next post.