- Algebras of Infinite Real Rank

We introduce the notion of weakly (strongly) infinite real rank for unital C∗-algebras. It is shown that a compact space X is weakly (strongly) infine-dimensional if and only if C(X) has weakly (strongly) infinite real rank. Some other properties of this concept are also investigated. In particular, we show that the group C∗-algebra C∗ (F∞) of the free group on countable number of generators has strongly infinite real rank. It is clear that some C-algebras of infinite real rank have infinite rank in a very strong sense of this word, while others do not. In order to formally distinguish these types of infinite ranks from each other we introduce the concept of weakly (strongly) infinite real rank. Proposition 1.1 characterizes usual real rank in terms of infinite sequences of self-adjoint elements and serves as a basis of our definition 2.1. We completely settle the commutative case by proving (Theorem 2.9) that the algebra C(X) has weakly infinite real rank if and only if X is a weakly infinite dimensional compactum. As expected, the group C-algebra C (F∞) of the free group on countable number of generators has strongly infinite real rank (Corollary 2.10). 1. Preliminaries All C-algebras below are assumed to be unital. The set of all self-adjoint elements of a C-algebra X is denoted by Xsa. The real rank of a unital C-algebra X, denoted by rr(X), is defined as follows [2]. We say that rr(X) ≤ n if for each (n+ 1)-tuple (x1, . . . , xn+1) of self-adjoint elements in X and every ǫ > 0, there exists an (n+1)-tuple (y1, . . . , yn+1) in Xsa such that ∑n+1 k=1 y 2 k is invertible and ∥ ∥ ∑n+1 k=1(xk − yk) 2 ∥