The Stone–Weierstrass problem for C*-algebras
A C$^*$-subalgebra $B$ of a C$^*$-algebra $A$ separates the pure states of $A$ together with $0$ if distinct pure states of $A$ have distinct restrictions to $B$ and no pure state of $A$ vanishes on $B$. For such $B$: when $A$ is commutative, $B = A$ by the Stone–Weierstrass theorem. Kaplansky proved $B = A$ when $A$ is of type I, and Glimm when $B$ moreover separates the states of $A$ in the weak$^*$ closure of the pure states. Sakai proved $B = A$ when $A$ is separable and there is a completely positive map $E : A \to B^{**}$ with $E(b) = b$ for $b \in B$, for instance when $A$ is separable and $B$ is nuclear. For separable $A$, Longo and Popa independently proved $B = A$ when $B$ moreover separates the factor states of $A$.
Problem. Let $A$ be a C$^*$-algebra and $B \subseteq A$ a C$^*$-subalgebra that separates the pure states of $A$ together with $0$. Must $B = A$?
Source: I. Kaplansky, The structure of certain operator algebras (1951); listed as O.12 in J. Peterson, Open problems in operator algebras (version of 15 May 2026).
Imported from math.uwaterloo.ca/~j37peter/problems.html.
Classification
- Primary
- OA — Operator Algebras
- Keywords
- C*-algebras, Peterson's operator algebra problems, pure states, Stone–Weierstrass theorem
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{kaplansky1951,
author = {Kaplansky, Irving},
title = {The structure of certain operator algebras},
journal = {Trans. Amer. Math. Soc.},
volume = {70},
number = {2},
pages = {219--255},
year = {1951},
doi = {10.1090/S0002-9947-1951-0042066-0},
}
@misc{peterson-o12,
author = {Peterson, Jesse},
title = {Open problems in operator algebras},
howpublished = {Web page},
year = {2026},
url = {https://www.math.uwaterloo.ca/~j37peter/problems.html},
note = {Problem O.12; version of 15 May 2026},
}
@article{glimm1960,
author = {Glimm, James},
title = {A {S}tone--{W}eierstrass theorem for {$C^*$}-algebras},
journal = {Ann. of Math.},
volume = {72},
number = {2},
pages = {216--244},
year = {1960},
doi = {10.2307/1970133},
}
@article{sakai1970,
author = {Sakai, Sh{\^o}ichir{\^o}},
title = {On the {S}tone--{W}eierstrass theorem of {$C^*$}-algebras},
journal = {Tohoku Math. J.},
volume = {22},
number = {2},
pages = {191--199},
year = {1970},
doi = {10.2748/tmj/1178242812},
}
@article{longo1984,
author = {Longo, Roberto},
title = {Solution of the factorial {S}tone--{W}eierstrass conjecture. {A}n application of the theory of standard split {$W^*$}-inclusions},
journal = {Invent. Math.},
volume = {76},
number = {1},
pages = {145--155},
year = {1984},
doi = {10.1007/BF01388497},
}
@article{popa1984,
author = {Popa, Sorin},
title = {Semiregular maximal abelian *-subalgebras and the solution to the factor state {S}tone--{W}eierstrass problem},
journal = {Invent. Math.},
volume = {76},
number = {1},
pages = {157--161},
year = {1984},
doi = {10.1007/BF01388498},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00962, "The Stone–Weierstrass problem for C*-algebras", clopen.solutions.