A Taiwanese investigation into the possible theft of chip technology at Taiwan Semiconductor Manufacturing Co. is putting a low-profile, lesser-known tech linchpin in Japan under unusual scrutiny.

Among the six people arrested by Taiwanese prosecutors for allegedly stealing trade secrets from the world’s largest contract chipmaker was a former employee at Tokyo Electron Ltd. Now, the Japanese company—one of the world’s biggest suppliers of chipmaking tools—is struggling to address the potential fallout with one of its most important customers and with governments in Tokyo and Taipei.

Tokyo Electron said that it fired an employee at its Taipei unit in connection with the case and is cooperating with the ongoing investigation. Its shares recouped some of the week’s losses Friday but re

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