Real rank (C*-algebras) explained

In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen.[1]

Definition

The real rank of a unital C*-algebra A is the smallest non-negative integer n, denoted RR(A), such that for every (n + 1)-tuple (x0, x1, ...,xn) of self-adjoint elements of A and every ε > 0, there exists an (n + 1)-tuple (y0, y1, ...,yn)of self-adjoint elements of A such that

n
\sum
i=0
2
y
i
is invertible and

\lVert

n
\sum
i=0

(xi-

2
y
i)

\rVert<\varepsilon

. If no such integer exists, then the real rank of A is infinite. The real rank of a non-unital C*-algebra is defined to be the real rank of its unitalization.

Comparisons with dimension

If X is a locally compact Hausdorff space, then RR(C0(X)) = dim(X), where dim is the Lebesgue covering dimension of X. As a result, real rank is considered a noncommutative generalization of dimension, but real rank can be rather different when compared to dimension. For example, most noncommutative tori have real rank zero, despite being a noncommutative version of the two-dimensional torus. For locally compact Hausdorff spaces, being zero-dimensional is equivalent to being totally disconnected. The analogous relationship fails for C*-algebras; while AF-algebras have real rank zero, the converse is false. Formulas that hold for dimension may not generalize for real rank. For example, Brown and Pedersen conjectured that RR(A &otimes; B) ≤ RR(A) + RR(B), since it is true that dim(X × Y) ≤ dim(X) + dim(Y). They proved a special case that if A is AF and B has real rank zero, then A &otimes; B has real rank zero. But in general their conjecture is false, there are C*-algebras A and B with real rank zero such that A &otimes; B has real rank greater than zero.[2]

Real rank zero

C*-algebras with real rank zero are of particular interest. By definition, a unital C*-algebra has real rank zero if and only if the invertible self-adjoint elements of A are dense in the self-adjoint elements of A. This condition is equivalent to the previously studied conditions:

This equivalence can be used to give many examples of C*-algebras with real rank zero including AW*-algebras, Bunce–Deddens algebras,[3] and von Neumann algebras. More broadly, simple unital purely infinite C*-algebras have real rank zero including the Cuntz algebras and Cuntz–Krieger algebras. Since simple graph C*-algebras are either AF or purely infinite, every simple graph C*-algebra has real rank zero.

Having real rank zero is a property closed under taking direct limits, hereditary C*-subalgebras, and strong Morita equivalence. In particular, if A has real rank zero, then Mn(A), the algebra of n × n matrices over A, has real rank zero for any integer n ≥ 1.

Notes and References

  1. Brown. Lawrence G. Pedersen. Lawrence G. Brown. Gert K. C*-algebras of real rank zero. Journal of Functional Analysis. July 1991. 99. 1. 131–149. 10.1016/0022-1236(91)90056-B. 0776.46026.
  2. Kodaka. Kazunori. Osaka. Hiroyuki. Real Rank of Tensor Products of C*-algebras. Proceedings of the American Mathematical Society. July 1995. 123. 7. 2213–2215. 10.1090/S0002-9939-1995-1264820-4. 0835.46053. free.
  3. Blackadar. Bruce. Kumjian. Alexander. Skew Products of Relations and the Structure of Simple C*-Algebras. Mathematische Zeitschrift. March 1985. 189. 1. 55–63. 10.1007/BF01246943. 0613.46049.