Yes, without the employee knowing. At some stage they will have to tell the employee if they intend to interview them or interview their friends/colleagues but there is nothing in law that states that any employer is required to inform an employee prior to investigating breaches of workplace conduct.
Employers do have the right to investigate about the employee's details or to even crosscheck any of the details that is mentioned by the employee. Though the employer would not be going deep into the details but would certainly be motivated to go if found some irrelevant information or some doubt about the accuracy of the details provided.
Conduct can be sufficiently "bad" in the eyes of an employer to allow for disciplinary steps to be cut. It is a matter of judgement and the severity of the misconduct needs to be justifiable before an employment tribunal.
Where do employees sit as far as their emails are concerned.
A relative has just been threatened with being sacked, after they had resigned, for gross professional misconduct for sending a contact to a friend about 4 months before. This friend had just recently left the company. The relative has been told they can either resign or face dismissal. They had already been escorted out of the company two days before on tendering their resignation. The company then went back through all their emails for 6 months to find out what had happened. They found an email to an ex employee giving a contact to this person. They then called her back in and she now has been told to resign with immediate effect losing 6 weeks pay or try and fight it and face disciplinary action.