Quantum spaces: notes and comments on a lecture by S. L. Woronowicz

Introduction. The concept of quantum spaces has been under study for some time now, and recently it has been attracting a growing amount of attention. This is in part due to the interest in quantum groups, which are considered a very promising concept by many, both physicists and mathematicians. It seems that quantum groups appear quite naturally in low-dimensional quantum field theory and statistical mechanics, they have also found important applications in the theory of knots. Other potential hoped-for applications include a possible role of quantum spaces in a future quantized theory of gravity. The theory of quantum spaces, and quantum groups in particular, is at present in a phase of rapid growth. One of the consequences of this is that the language used in the field still remains somewhat ambiguous, even concerning the definition of basic concepts. Here, the category of locally compact quantum spaces will be