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I am afraid that the answer is yes. If the computer that the employee is using belongs to the employer, the employer can monitor its use, even if the employee is using a non-work email account. This has been held up in numerous court opinions in both state and federal court, stemming from the fact that if the employee is using the internet and equipment provided by the employer, the employer has every right to monitor its use.
There is no mandate that the employee be warned beforehand about the monitoring. The employer can choose to warn the employee, but this is at their discretion; there is no law that they have to do so.
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