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.
[1] 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
If no such integer exists, then the real rank of A is infinite.
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.
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.
Having real rank zero is a property closed under taking direct limits, hereditary C*-subalgebras, and strong Morita equivalence.