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Purely Infinite C^*-Algebra

A purely infinite C^*-algebra, in the nonsimple comparison sense, is a C^*-algebra in which every positive element h is properly infinite in stable Cuntz comparison:

h direct sum h≾h.
(1)

Zero is included in this convention. Weak pure infiniteness in the fixed-amplification sense requires one integer n>=1 such that h^( direct sum n) is properly infinite for every positive h. Strong pure infiniteness requires, for all positive elements x,y and epsilon>0, coefficients s,t in the algebra such that

||s^*x^2s-x^2||<epsilon,
(2)
||t^*y^2t-y^2||<epsilon,
(3)

and

||s^*xyt||<epsilon.
(4)

See also

Cuntz Algebra, Cuntz Semigroup, Nuclear C-Star-Algebra, Stably Finite C-Star-Algebra

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References

Kirchberg, E. and Rørdam, M. "Infinite Non-Simple C^*-Algebras: Absorbing the Cuntz Algebra O_infty." Adv. Math. 167, 195-264, 2002. https://doi.org/10.1006/aima.2001.2041.Kirchberg, E. and Sierakowski, A. "Strong Pure Infiniteness of Crossed Products." Ergodic Th. Dynam. Sys. 38, 220-243, 2018. https://doi.org/10.1017/etds.2016.25.

Cite this as:

Weisstein, Eric W. "Purely Infinite C^*-Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PurelyInfiniteC-Star-Algebra.html

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