Naimark

Mark Naimark

Mark Aronovich Naimark was a Soviet mathematician.He was born in Odessa, Russian Empire into a Jewish family and died in Moscow, USSR...

's problem is a question in functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

. It asks whether every C*-algebra that has only one irreducible representation up to unitary equivalence is isomorphic to the algebra of compact operator

Compact operator on Hilbert space

In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...

s on some Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

.

The problem was solved in the affirmative for special cases (separable and type I C*-algebras). used ◊ to construct a C^*-algebra serving as a counterexample to Naimark's problem and showed that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ₁ elements" is independent of the axioms of Zermelo-Fraenkel set theory and the axiom of choice (ZFC).