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

C*-algebraic extensions and BDF-Theory

Let us consider a metrizable compact topological space Omega, and let us consider the classification problem for all possible short exact sequences of C*-algebras of the form

bdf-sequence

where K is the ideal of compact operators in a separable Hilbert space. It coincides with the commutant (the ideal generated by all possible commutators) of Xi in other words Omega is the space of characters of Xi. It can be shown [BDF] that homotopy classes of such extensions are in one-to-one correspondence with the elements of the first K-homology group of Omega. In other words, described non-commutative extensions reflect the topology of Omega. It is also of some interest to consider more general situations, where K is replaced by a different C*-algebra. For a general introduction to C*-algebraic extensions we refer to [We].

As a paradigmic example of a non-trivial extension of the described type, let us mention the extension generated by the shift operator T acting in 2-summable-sequences. By definition, this operator is defined by

shift-operator

where the vectors e-n form the canonical basis in H. The algebra Xi is defined as the C*-algebra generated by T. We have omega-u(1), as characters of Xi are completely determined by their values on T, and the values cover the unit circle U(1). This extension plays a central role in a very elegant proof [We] of the Bott periodicity for general C*-algebras.

It is worth mentioning that extensions of the described type naturally appear in C*-algebraic foundations of a causal subquantum mechanics [D3] where Xi plays the role of the algebra of subquantum variables and Omega is the subquantum space of a given physical system.

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