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The problem of zero divisors in convolution algebras of supersolvable Lie groups

We prove a variant of the Titchmarsh convolution theorem for simply connected supersolvable Lie groups, namely we show that the convolution algebras of compactly supported continuous functions and compactly supported finite measures on such groups do not contain zero divisors. This can be also viewed as a topological version of the zero divisor conjecture of Kaplansky.