Real rank of extensions of \(C^{*}\)-algebras (Q6581187)

As mentioned in the abstract of this paper, the Brown-Pedersen real rank (RR) of a \(C^*\)-algebra with a closed ideal is estimated by the real ranks of the closed ideal and its quotient \(C^*\)-algebra, in the following four cases of (1) \(C^*\)-algebras with RR zero and Rieffel stable rank one, and K-theory \(K_1\)-group vanishing (or zero), (2) simple \(C^*\)-algebras of being purely infinite, (3) simple \(C^*\)-algebras with RR zero of absorbing the Jiang-Su algebra as a tensor product, and (4) separable \(C^*\)-algebras with approximate units of projections and the corona factorization property, of absorbing the \(C^*\)-algebra of compact operators as a tensor product.\N\NIntroduced as a notion is the extension real rank of a \(C^*\)-algebra (universal extension), which may be denoted as exRR if you like, which involves self-adjoint elements of the multiplier algebra \(M(A)\) of \(A\) (non-simple).