The Real Rank of CCR C*-Algebra

Abstract

Abstract. We estimate the real rank of CCR C ∗ -algebras under some assumptions. Asapplications we determine the real rank of the reduced group C ∗ -algebras of non-compactconnected, semi-simple and reductive Lie groups and that of the group C ∗ -algebras ofconnected nilpotent Lie groups. 1. IntroductionThe real rank for C ∗ -algebras was introduced by Brown and Pedersen [3]. Bydefinition, we say that a unital C ∗ -algebra A has the real rank n = RR(A) if n isthe smallest non-negative integer such that for any e > 0 given, any self-adjointelement (a j ) n+1j=1 ∈ A n+1 with a j = a ∗j is approximated by a self-adjoint element(b j ) n+1j=1 ∈ A n with b j = b ∗j such that ka j − b j k < e (1 ≤ j ≤ n + 1) andP +1=1 b 2j is invertible in A. For a non-unital C ∗ -algebra, its real rank is defined by that ofits unitization by C. By definition, RR(A) ∈ {0,1,2,··· ,∞}.On the other hand, CCR C ∗ -algebras are very well known in the C -algebratheory such as the representation theory and structure theory of C