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Extended Trace Cone

The extended trace cone T(A) of a C^*-algebra consists of all lower semicontinuous maps tau:A_+->[0,infty] that vanish at zero, are additive and positively homogeneous, and satisfy tau(x^*x)=tau(xx^*). The finite domain is not required to be dense. Its canonical topology is characterized by

tau_i->tau<==>limsup_(i)tau_i((a-epsilon)_+)<=tau(a)<=liminf_(i)tau_i(a),

where a in A_+ and epsilon>0. The positive cut (a-epsilon)_+ uses continuous functional calculus. Every closed two-sided ideal I contributes a trace taking value zero on I_+ and infinity on positive elements outside I. Thus the cone retains ideal information in addition to finite trace values.

See also

Cuntz Semigroup, Nuclear C-Star-Algebra, Razak-Jacelon Algebra, Tracial State

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References

Schafhauser, C.; Tikuisis, A.; and White, S. "Nuclear C^*-Algebras: 99 Problems." 12 Jun 2025. https://arxiv.org/abs/2506.10902.

Cite this as:

Weisstein, Eric W. "Extended Trace Cone." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExtendedTraceCone.html

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