After an enormous amount of work by many authors (perhaps two thousand pages), it has been shown that every unital simple separable C*-algebra satisfying certain natural hypotheses (finite nuclear dimension, and the Universal Coefficient Theorem---the latter possibly redundant) is a limit of type I C*-algebras, and that these can be classified by simple invariants. The non-unital case looks promising, but may still require a considerable amount of work.

I will first discuss

the classic Sobolev type inequalities and also the logarithmic Sobolev inequality and will compare them.

Moreover, one of my main results, which is the logarithmic Sobolev inequality on the noncommutative

two torus (NCT2), will be be proved for a class of elements in NCT2. Afterwards, I will introduce a family of

spectral triples, each of which represents a class of conformaly perturbed metrics on the noncommutative

three torus (NCT3) as a noncommutative compact spin manifold. Then using Connes' pseudodifferential

calculus, I will define the scalar curvature of NCT3 equipped with the mentioned conformaly perturbed

metrics and will compute it by an explicit formula.