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Cuntz Semigroup

The Cuntz semigroup Cu(A) of a C^*-algebra is the ordered semigroup of equivalence classes of positive elements of A tensor K, where K is the compact operators on an infinite-dimensional separable Hilbert space. The comparison relation is

a≾b<==>v_n^*bv_n->a in norm for some v_n in A tensor K.

Equivalence means comparison in both directions, order is induced by comparison, and addition is given by orthogonal direct sum. The semigroup also retains increasing sequential suprema and its way-below relation. For A=C, its elements are ranks 0,1,2,...,infty. A functional on this semigroup preserves zero, addition, order, and increasing suprema and takes values in [0,infty]. For an extended lower semicontinuous trace tau, the corresponding functional is [a]|->d_tau(a)=lim_(n)tau(a^(1/n)) with unnormalized matrix trace.

See also

C*-Algebra, Extended Trace Cone, Strict Comparison

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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. "Cuntz Semigroup." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CuntzSemigroup.html

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