A simpleC*-algebra with a finite and an infinite projection

By M. Rørdam. 2002

167 citations recorded by the source corpus.

An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a3125fd5b9a9a94634dccc0941af0e7079c6ea74

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Citations in the corpus

Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “Remarkably, there exist algebras satisfying the hypotheses of the above theorem which are not Z-stable ([51], [56], [57]).”
Central Sequences in C*-Algebras and Strongly Purely Infinite Algebras (2006, background): “Rørdam [30]).”
Divisibility properties for C*‐algebras (2011, unknown): “Following the notation in [26], for each set I = {i1, i2, .”
Recasting the Elliott conjecture (2006, background): “1) to be insufficient for the classification of all simple, separable, and nuclear C∗-algebras ([30], [34], [35]).”
Subshifts and perforation (2010, background): “Over the last several years, however, the classification program has had to come to terms with examples of Villadsen [29, 30], Rørdam [22], and Toms [26] that have pointed toward the need for regularity assumptions like Z-stability or the use of invariants of a finer type [8].”
Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, background): “Rørdam exhibited a unital separable simple C*-algebra which is finite but not stably finite ([86]) However, it was A Toms ([91]) who showed that there are unital simple AH-algebras”
Completeness of the isomorphism problem for separable C*-algebras (2013, background): “real rank zero algebras, AH algebras of slow dimension growth or separable, simple, purely infinite, nuclear algebras (modulo the universal coefficient theorem) [18,21–23,49,50,64] and there have been dramatic breakthroughs in the program, including the counterexamples to the general classification ”
Leavitt path algebras of separated graphs (2010, background): “It is worth to mention that Rørdam has constructed in [37] examples of simple C-algebras having both finite and (properly) infinite projections.”
Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “If one can arrange that u0 = 1B and hence (ut ∈ U0(B) for all t), then we say that ϕ and ψ are strongly asymptotically unitarily equivalent, written ϕ ≈sasu ψ. 1.2 The concept of strongly self-absorbing C∗-algebras was formally introduced in [54, Definition 1.3]: Definition: (i) A separable unital…”
The nuclear dimension of C*-algebras (2009, methodology): “In [23], Rørdam constructed a simple, separable, unital, and nuclear C-algebra containing a finite and an infinite projection.”
Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra (2012, background): “In the quest to classify simple separable nuclear C∗-algebras, pioneered by George Elliott, it has become necessary to invoke some regularity property of the C∗-algebra, without which classification—at least by simple K-theoretical invariants—is known to be impossible, see for example [24] and [18].”
A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, background): “Rørdam exhibited an amenable separable simple C∗-algebra which is finite but not stably finite ([97]).”