Assume that M is a compact Riemannian manifold and E is the complexification of TM. Let A be the complex C* algebra H0m(E,E) with natural operations.
Are there some reference devoted to computation of K-theory and also topological stable rank of A.? Is this rank finite?Does it depend only on dimension of M?
BTW: I remember that I found in a "Fields institute monograph" a statment that the COMPLETION of A is a C* algebra.(I do not remember the serial number of that monograph). But is not "COMPLETION", unnecessary?I mean is not A, already complete?
The hypotheses here are overly restrictive; the topological stable rank is known for any n-homogeneous algebra A over a compact space M (which is indeed already complete). For n=1, you have C(M) which has tsr(A) = 1 + [dim(M)/2]. For n>1 the formula is {[dim(M)/2]/m} + 1. (Here [ ] and { } are the "floor" and "ceiling" functions.) See Rieffel, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333.
Dear Prof. Rosenberg
Thank you very much for your answer. Assume that all irreducible representation of a C* algebra have a fix dimension n. Is it true to say that it is in the form of endomorphism algebra of a n-dimensional vector bundle?
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