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Topological generation results for free unitary and orthogonal groups. (English) Zbl 1516.20116

Summary: We show that for every \(N \geq 3\) the free unitary group \(U_N^+\) is topologically generated by its classical counterpart \(U_N\) and the lower-rank \(U_{N - 1}^+\). This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all \(N \neq 3\), also hold at \(N = 3\). Specifically, all discrete quantum duals \(\widehat{U_N^+}\) and \(\widehat{O_N^+}\) are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, \(U_N^+\) are topologically generated by \(U_N\) and their maximal tori \(\widehat{\mathbb Z^{\ast N}}\) (dual to the free groups on \(N\) generators) and similarly, \(O_N^+\) are topologically generated by \(O_N\) and their tori \(\widehat{\mathbb Z_2^{\ast N}}\).

MSC:

20G42 Quantum groups (quantized function algebras) and their representations
16T20 Ring-theoretic aspects of quantum groups
46L52 Noncommutative function spaces
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