A Brief Introduction to Quantum Geometry

Table of Contents

Introduction
From Geometry to Algebra: 1
Part 2
Non-commutative Generalizations
Examples
Spaces With Indistinguishable Points
Quantum Groups and Quantum Bundles
Quantum Tori and Matrix Spaces
Supermanifolds
Quantization of Symplectic Manifolds
C*-Algebraic Extensions and BDF-Theory
Literature

Reformulating Basic Geometrical Concepts [1/2]

We shall now explain how some of the most important geometrical concepts are translated into the language of algebra.

Points

Let us assume that X is a compact topological space, and let A=C(X) be the *-algebra of continuous complex-valued functions on X. The algebraic operations in A are the standard multiplication and addition of functions, and the *-operation is the standard complex conjugation.

Every element x of X naturally gives rise to a linear functional linear-functional-x defined by definition-of-kappax. This map is multiplicative in the sense that

multiplicativity
for each fg-in-algebra, and hermitian in the sense that
hermicity-property.
It is also non-zero. In other words kappa is a character on A.

Conversely, let us consider an arbitrary character kappa-algebra-2-c. Then it can be shown that there exists a unique point point-x such that kappa-kappax. In other words, we have a natural bijection between points of X and characters of A. It is worth noticing that this characterization of points remains valid at the smooth level, too. In this case X is a smooth manifold and the *-algebra consists of smooth functions.

Gelfand-Naimark Theorem

The algebra A=C(X) of complex-valued continuous functions on a compact topological space X, equipped with the maximum norm
definition-maximum-norm
is a commutative C*-algebra. The classical theorem of Gelfand and Naimark characterizes the algebras of the form A=C(X), as commutative unital C*-algebras. In other words, for every commutative unital C*-algebra A there exists (up to the homeomorphisms) a unique (up to homeomorphisms) compact topological space X such that A is isomorphic to C(X).

As we have just mentioned, the points of the space X are recovered as characters of the associated algebra A. In terms of this identification, the topology on X coincides with the *-weak topology, induced from the dual space A* (by definition continuous linear functionals, it turns out that homomorphisms between C*-algebras are automatically continuous, in particular characters are continuous linear functionals).

The theory of compact topological spaces is the same as the theory of commutative unital C*-algebras. If we relax the unitality assumption (dealing with arbitrary commutative C*-algebras), then the category of corresponding spaces is enlarged to the level of locally-compact spaces. If X is non-compact then A is consisting of continuous functions on X that vanish at infinity.

If X is a measurable space (without any extra structure) then the relevant *-algebra is consisting of all essentially bounded measurable functions on X. It becomes a (commutative) von Neumann algebra, if equipped with the essential supremum norm. It can be shown that every commutative von Neumann algebra is of this form. The entire measure theory is essentially the same as the theory of commutative von Neumann algebras.

Continuous Maps and Direct Products

Now let us consider two compact topological spaces X and Y, and let us denote by A and B the *-algebras of continuous functions over X and Y respectively. Let topological-map be an arbitrary continuous map between X and Y. To this map, we can associate another map associated-map defined via the composition pull-back-def It is easy to see that the map lambda is a unital *-homomorphism between B and A.

Conversely, let us consider an arbitrary unital *-homomorphism homomorphism-B2A. Then it can be shown that lambda is always of the form uniqueness-of-F for a uniquely determined continuous map topological-map. In other words we have a natural bijection between continuous maps from X to Y, and unital *-homomorphisms from B to A. The same algebraic characterization holds at the smooth level, too (smooth maps between compact smooth manifolds are in a one-to-one correspondence with unital *-homomorphisms between the associated algebras of smooth functions).

Properties of the map F are reflected as properties of lambda_F and vice versa. For example, F is surjective if and only if lambda_F is injective, and F will be injective if and only of lambda_F is surjective. If C is the C*-algebra of continuous functions on the direct product X×Y then the following natural identification holds:

tensor-product-identification
where the product tensor-product-c* here is the C*-algebraic tensor product.

Symmetry

Symmetry transformations of the space X can be understood as certain homeomorphisms of X. In accordance with the previous paragraph, homeomorphisms of X are in one-to-one correspondence with automorphisms of A. The same holds at the smooth level: If X is a smooth manifold then diffeomorphisms of X are in a natural bijection with automorphisms of the corresponding algebra of smooth functions.

Probability Measures

Let us assume that X is a metrizable compact topological space (metrizability of X is equivalent to the separability of A=C(X) in its uniform norm). Let us consider a probability measure mu defined on the sigma-field B(X) of Borel subsets of X. Let rho-mu be a linear functional defined as the Lebesgue intergral

def-of-rho-mu

The map rho is linear, normalized ( unitality-rho) and positive (the values of rho on non-negative functions are non-negative numbers) functional on A.

Conversely, if rho-generic is an arbitrary positive and normalized linear functional on A then there exists a unique probability measure probability-measure-borels such that rho-recovered. This is the classical Riesz representation theorem, establishing a natural correspondence between probability measures on X and positive normalized functionals on A.

Compact Topological Groups

Let X be a compact topological group. This means that X is a compact topological space equiped with a group structure, such that the product map group-structure is continuous (it can be shown that in the compact case continuity of the product implies continuity of the inverse map). At the dual level, the product map is represented by a *-homomorphism coproduct-map. The associativity property of the product is equivalent to the coassociativity property
coass-property

It can be shown that the remaining two group axioms (the existence of the neutral element and the existence of the inverse elements) are equivalent to a single assumption that two linear subspaces spanned by all the elements of the form left-module-a-phi-b and right-module-phi-ab respectively, are both everywhere dense in tensorial-square.

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