Buddy says: If you have not received a verbal or written resignation from the employee, then you will need to ascertain if the employee has left the company or simply gone AWOL.  The latter will be grounds for potential dismissal on the grounds of misconduct (i.e. unauthorised absence) and will need to be dealt with in accordance with company procedures as well as the statutory minimum disciplinary & dismissal procedure. 

Employer Information:   The first step would be to write to the employee, sending this by recorded delivery so you have evidence of postage.  This letter should state that as the employee has not turned up to work, you are assuming that he has decided to resign and no longer wishes be bound by his terms and conditions of employment.  Ask the employee to respond within 7 days, to confirm the position.  If you do not receive a response, simply write to the employee again accepting his resignation with effect from his last working day.  You should enclose his P45 and any pay owing up to and including his last working day (along with any other contractual pay/benefits owing).  

Taking steps to investigate the facts surrounding the resignation and allowing a reasonable cooling off period is advisable.  For example, had the employee been in trouble at work, was he being bullied/harassed etc.  If something untoward does come to light, you can respond to this in the most appropriate way (i.e. is there a grievance that needs dealing with, is the employe likely to allege discrimination or constructive unfair dismissal?). 

Note, in a claim for unfair dismissal, the onus is always on the employee to prove that they had been dismissed.  They will likely fail in proving a dismissal if you follow the above steps and don't act rashly.