Intrinsic covering dimension for nuclear \(C^*\)-algebras with real rank zero
It is shown that the Winter-Zacharias nuclear dimension \((n)\) [\textit{W. Winter} and \textit{J. Zacharias}, Adv. Math. 224, No. 2, 461--498 (2010; Zbl 1201.46056)] and the Kirchberg-Winter decomposition rank \((n)\) [\textit{E. Kirchberg} and \textit{W. Winter}, Int. J. Math. 15, No. 1, 63--85 (2004; Zbl 1065.46053)] for a \(C^*\)-algebra can be characterized by a finite number \((n+1)\) of finite-dimensional \(C^*\)-subalgebras and the several approximations by their positive elements in the closed unit ball as the (locally approximate) covering dimensions for a \(C^*\)-algebra, or the intrinsic nuclear dimension and the intrinsic decomposition rank for a \(C^*\)-algebra, under the condition of the \(C^*\)-algebra to have Brown-Pedersen real rank zero [\textit{L. G. Brown} and \textit{G. K. Pedersen}, J. Funct. Anal. 99, No. 1, 131--149 (1991; Zbl 0776.46026)]. As a corollary, a \(C^*\)-algebra with intrinsic decomposition rank finite becomes a Popa algebra [\textit{S. Popa}, Pac. J. Math. 181, No. 1, 141--158 (1997; Zbl 0907.46046)]. As another corollary, a simple unital \(C^*\)-algebra with real rank zero and decomposition rank finite and a unique tracial state becomes tracially AF [\textit{H. Lin}, Trans. Am. Math. Soc. 353, No. 2, 693--722 (2001; Zbl 0964.46044)]. Under the same condition of real rank zero, it is shown that being nuclear for a \(C^*\)-algebra can be characterised by finite numbers of finite-dimensional \(C^*\)-subalgebras and the several approximations by their positive elements in the closed unit ball, together with completely positive, contractive maps from the \(C^*\)-algebra to the intersections of the finite-dimensional \(C^*\)-algebras with the commutants of the positive elements [\textit{I. Hirshberg} et al., Adv. Math. 230, No. 3, 1029--1039 (2012; Zbl 1256.46019)].
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