How many non isomorphic finite dimensional $C^*$ algebras if the dimensions without a bound? Is it countable or uncountable?
2026-03-26 10:07:07.1774519627
non isomorphic finite dimensional $C*$ algebras
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For each $n$, the set of (isomorphism classes) of $C^*$-algebras of dimension $\leq n$ is finite (being the number of collections $\{(n_1,m_1),\ldots,(n_k,m_k)\}$ of pairs of natural numbers with the $m_j$ distinct such that $n_1\cdot m_1^2+\cdots+n_k\cdot m_k^2\leq n$), hence the collection of (isomorphism classes) of finite-dimensional $C^*$-algebras is countable.