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This paper studies the tracial stability of C*-algebras, which is a general property of stability of relations in a Hilbert-Schmidt-type norm defined by a trace on a C*-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear C*-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory. [ABSTRACT FROM AUTHOR]

In his paper mentioned in the title, which appears in the same issue of this journal, Mehdi Radjabalipour derives the cyclic decomposition of an algebraic linear transformation. A more general structure theory for linear transformations appears in Irving Kaplansky's lovely 1954 book on infinite abelian groups. We present a translation of Kaplansky's results for abelian groups into the terminology of linear transformations. We add an additional translation of a ring-theoretic result to give a characterization of algebraically hyporeflexive transformations and the strict closure of the set of polynomials in a transformation T. [ABSTRACT FROM AUTHOR]

Suppose π : A → B is a surjective unital *-homomorphism between C*-algebras A and B, and 0 ≤ α ≤ 1 with a € A. We give a sufficient condition that ensures there is a proection p € A such that π (p) = π (a). An easy consequence is a result of [L. G. Brown and G. k. Pedersen, C*-algebras of real rank zero, J. Funct. Anal. 99 (1991) 131--149] that such a p exists when A has real rank zero. [ABSTRACT FROM AUTHOR]