Bounded Rank of C ∗-algebras
نویسنده
چکیده
We introduce a new concept of the bounded rank (with respect to a positive constant) for unital C∗-algebras as a modification of the usual real rank. We present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K > 0 there exists a separable unital C∗-algebra Z n such that every other separable unital C∗-algebra of bounded rank with respect to K at most n is a quotient of Z n . We also introduce the notion of weakly (strongly) infinite real (bounded) rank for unital C∗-algebras as a tool for distinguishing various types of C∗-algebras of infinite real (bounded) rank. In the commutative case we prove that a unital C∗-algebra has a weakly infinite real (bounded) rank if and only if its spectrum is weakly infinite-dimensional in the standard topological sense. We also show that C∗ (F∞) not only doesn’t have finite real rank, but actually has strongly infinite bounded rank.
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