Abstract
Linear maps acting on Hilbert spaces play an important role in quantum mechanics. Such a linear map is called an operator. Well-known examples are the Pauli matrix σ z , which measures the spin in the z direction, and the position and momentum operators P and Q of a single particle on the line. The first acts on the Hilbert space \(\mathbb{C}^{2}\), while the latter are defined on dense subspaces of \(L^{2}(\mathbb{R})\). An operator algebra is an algebra of such operators, usually with additional conditions, such as being closed in a certain topology. Here we introduce some of the basic concepts in the theory of operator algebras. The material here is standard, and by now there is a huge body of textbooks on the subject, most of which cover a substantially bigger part of the field than these notes. Particularly recommended are the two volumes by Bratteli and Robinson (Operator algebras and quantum statistical mechanics, vol. 1, 2nd edn. Texts and monographs in physics. Springer, New York, 1987; Operator algebras and quantum statistical mechanics, vol. 2, 2nd edn. Texts and monographs in physics. Springer, Berlin, 1997), which contain many applications to physics. Many of the topics covered here are studied there in extenso. The books by Kadison and Ringrose provide a very thorough introduction to operator algebras, and contain many exercises (Kadison and Ringrose, Fundamentals of the theory of operator algebras. Vol. II: advanced theory. Graduate studies in mathematics, vol. 16. American Mathematical Society, Providence, RI, 1997; Kadison and Ringrose, Fundamentals of the theory of operator algebras, Vol. I: elementary theory. Pure and applied mathematics, vol. 100. Academic, New York, 1983). The book by Takesaki (Theory of operator algebras. I. Encyclopaedia of mathematical sciences, vol. 124. Springer, Berlin, 2002) [and the subsequent volumes II and III (Takesaki, M.: Theory of operator algebras. II. Encyclopaedia of mathematical sciences, vol. 125. Springer, Berlin, 2003; Theory of operator algebras. III. Encyclopaedia of mathematical sciences, vol. 127. Springer, Berlin, 2003)] is a classic, but is more technical. Volume I of Reed and Simon’s Methods of Modern Mathematical Physics (Reed, M., Simon, B.: Methods of modern mathematical physics. I: functional analysis, 2nd edn. Academic [Harcourt Brace Jovanovich, Publishers], New York, 1980) or Pedersen’s Analysis Now (Pedersen, Analysis now. Graduate texts in mathematics, vol. 118. Springer, New York, 1989) cover the necessary tools of functional analysis (and much more), but do not cover most of the material on operator algebras we present here.
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Notes
- 1.
One can also define Hilbert spaces over \(\mathbb{R}\), but these will play no role in these lectures.
- 2.
In the mathematics literature the inner product is often linear in the first variable, and anti-linear in the second. We adopt the physics conventions.
- 3.
The appropriate setting for a mathematical theory of integration is measure theory. We will briefly come back to this when we discuss the functional calculus, but for the most part of this book it will play no role. Details can be found in most graduate level texts on analysis or functional analysis.
- 4.
In abstract nonsense language: one has to say in which category one is working.
- 5.
The proof requires the Riesz representation theorem. First note that the map \(\xi \mapsto \overline{\langle A\xi,\psi \rangle }\) is a bounded linear functional (that is, a linear map to \(\mathbb{C})\) on \(\mathcal{H}\). By the Riesz representation theorem there is a \(\eta \in \mathcal{H}\) such that \(\langle \eta,\xi \rangle = \overline{\langle A\xi,\psi \rangle }\) for all \(\xi \in \mathcal{H}\). Set A ∗ ψ = η. This uniquely defines a bounded operator A ∗ with the right properties.
- 6.
This is essentially how the real numbers are usually defined. Namely, they are limits of Cauchy sequences of rational numbers.
- 7.
We will deal almost exclusively with vector spaces over \(\mathbb{C}\).
- 8.
If X is a topological space, the Borel measurable sets \(\mathcal{B}(X)\) is the smallest collection of subsets of X such that \(O \in \mathcal{B}(X)\) for all open sets O and \(\mathcal{B}(X)\) is closed under the complement operation and under taking countable unions. The elements of \(\mathcal{B}(X)\) are called Borel sets. A Borel measure μ assigns a positive real number μ(X) to each of the Borel sets in a way compatible with the structure of the Borel sets. For example, μ(X 1 ∪ X 2) = μ(X 1) +μ(X 2) for two disjoint Borel sets X 1 and X 2. Intuitively speaking, μ(X) tells us how “big” the set X is. Once one has a measure, it is possible to define integration with respect to that measure.
- 9.
One might hope that the operators actually converge in norm, but in general this is too much to ask for.
- 10.
Some authors label the matrices by σ 1, σ 2, σ 3 and write σ 0 for the identity matrix. Later on we will also use a superscript instead of a subscript, since we will also need to index the site on which the matrix acts.
- 11.
The proof of this, which is not too difficult, relies on some facts that we have not discussed here. See for example [6, Theorem 4.3.4(iv)]. One can even show that ω can be chosen to be a pure state.
- 12.
A proper treatment would require discussion of so-called normal states and of von Neumann algebras. Since we will not need these concepts later on, we will not go into the details here.
- 13.
This excludes unbounded operators. One can argue that this is no severe objection, since in actual experiments there will always be a bounded range in which a measurement apparatus can operate. If one considers an unbounded operator on a Hilbert space, one can show that its spectral projections are contained in a von Neumann algebra (a subclass of the C ∗-algebras). These spectral projections essentially project on the subspace of states with possible measurement outcomes in a bounded range. The advantage of working with bounded operators is that they are technically much easier to handle. Nevertheless, one can incorporate unbounded operators in the framework (by passing to the GNS representation of a state, for example). We will encounter some examples of this later on.
- 14.
We changed the notation a bit to emphasize the dependence on A.
- 15.
Note that unitary equivalence already appeared in the uniqueness statement of the GNS representation.
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Naaijkens, P. (2017). Operator Algebras. In: Quantum Spin Systems on Infinite Lattices. Lecture Notes in Physics, vol 933. Springer, Cham. https://doi.org/10.1007/978-3-319-51458-1_2
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