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C^*-Algebra

A C^*-algebra is a Banach algebra with an antiautomorphic involution * which satisfies

(x^*)^*=x
(1)
x^*y^*=(yx)^*
(2)
x^*+y^*=(x+y)^*
(3)
(cx)^*=c^_x^*,
(4)

where c^_ is the complex conjugate of c, and whose norm satisfies

||xx^*||=||x||^2.
(5)

An operator algebra weight may take the value infinity on positive elements. Blecher and Choi (2026) proved that its lower semicontinuity, and in the von Neumann case its normality, can be tested on commutative subalgebras. Blecher and Choi (2026) report AI assistance with the mathematical proof.

See also

C*-Algebra Character, C*-Algebra Isometry, C*-Algebra Tensor Product, Cuntz Algebra, Dimension-Drop Algebra, Jiang-Su Algebra, K-Theory, Kadison-Kastler Conjecture, Kadison's Similarity Problem, Nuclear C*-Algebra, Operator System, Pre-C*-Algebra, Quasitrace, Stably Finite C*-Algebra, Tracial State

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References

Blecher, D. P. and Choi, I. "A Solution to Haagerup's Problem: Normality of Weights is Commutatively Determined." 26 Oct 2026. https://arxiv.org/abs/2610.35774.Cuntz, J. and Echterhoff, S. (Eds.). C* Algebras: Proceedings of the SFB-Workshop on C* Algebras, Münster, Germany, March 8-12, 1999. Berlin, Germany: Springer-Verlag, 2000.Davidson, K. R. C*-Algebras by Example. Providence, RI: Amer. Math. Soc., 1996.Wegge-Olsen, N. E. K-Theory and C*-Algebras: A Friendly Approach. Oxford, England: Oxford University Press, 1993.

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C^*-Algebra

Cite this as:

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

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