One Noncommutative Differential Calculus Coming from the Inner Derivation

We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain C∗-algebra, this calculus is closely related to the classical one when the algebra associates a deformation parameter. Introduction Noncommutative geometry is developed by Alain Connes. In his work [8], the cyclic cohomology, which is considered to be the classical differential calculus counterpart, is introduced. In this paper, we are dealing with one new noncommutative differential calculus. When considering some C-algebra Aθ which is parametrized by a number θ ({Aθ} is a family of continuous field of C-algebra, when θ = 0, A0 is commutative, for instance, quantum tori, quantum plane etc), this differential calculus will be deforming to the classical one when θ goes to 0. The important point to note here is that the definition is trivial if the underlying algebra is commutative. When combining the condition (2) of the following definition from Rieffel [12], it is very easy to see that applying this noncommutative differential calculus for the strict deformation quantization algebra, it will be deforming to the classical differential calculus. Definiton [12] 0.1. A strict deformation quantization of A in the direction of Λ means an open interval I of real numbers containing 0, together with, for each ~ ∈ I, an associative product ∗~, an involution ∗~ , and a C-norm || · ||~ on A, which for ~ = 0 are the original pointwise product, complex conjugation involution, and supremum norm, such that (1) for every f ∈ A, the function ~ 7→ ||f ||~is continuous (2) for every f, g ∈ A, ||(f ∗~ g − g ∗~ f)/i~− {f, g}||~ converges to 0 as ~ goes to 0. The notation Λ means a skew 2-vector field and {, } is a Poisson bracket. For a full treatment of the strict deformation quantization theory, we refer the reader to [12]. This paper is organized as follows. In the first section, we define carefully this new noncommutative differential calculus and its basic properties. In the second section, we proceed to the study of the Dirichlet form constructed from the inner derivation. We will show this Dirichlet form is symmetric, Markov, conservative, completely positive and strongly local. In section 3, 4, 5, 6, 7, we apply this calculus for Date: February 2, 2008.