Say someone steals from a store. This is recorded and reported by an employee, but the owner doesn't care.
Will police only arrest the perpetrator if the victim cares? Is it necessary for the victim to care about being a victim?
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You tagged your answer "California". It depends on jurisdiction.
In Italy we the answer can be split in two:
Crimes for what is necessary a complaint to the police by the victim. This is called "querela".
By "querela", the victim informs the police of the crime, provide evidence, and formally ask the prosecutor to conduct an investigation, and formally asks the prosecutor to fill charges against the perpetrator.
The prosecutor must fill charges. This is called "azione penale obbligatoria" (compulsory penal action by the prosecutor). But, the prosecutor will need to ask a "first instance judge" (called "GIP") to either proceed to a formal trial or to archive the accusations.
In either case the judge can do whatever he/she wants. He can accept the prosecutor request to proceed to a trial, he can accept the prosecutor request to archive the accusation, he can deny the prosecutor request to proceed to a trial and order for the accusations to be archived, he can deny the prosecutor request for the accusation to be archived and order a trial (although this latter case is not very common).
In this kind of cases, if "querela" is missing (the formal complaint by the victim to the police), then there is no trial.
If the victim withdraws "querela", the trial must stop.
Some crimes do not require "querela". For example homicide or rape (but also some other petty crimes). In these cases the prosecution will move forward also without a formal request by the victim to prosecute the crime.
Police have discretion whether to arrest and prosecute or not.
In exercising this discretion they may take into account whether victim is identified and what he/she says, but there are no legally enforceable rules around it. Discretion is discretionary after all.
Other factors that may influence the police's decision whether to arrest notwithstanding whether the victim cares may include but are not limited to:
Will police only arrest if the victim cares?
No. The police will not only arrest, but also will secure evidence and conduct other activities that will be useful to the prosecutor. The police might also make arrangements to facilitate that everything in the crime scene goes back to normal.
Note that your first paragraph is contradictory. Stealing from a store means that the victim is the owner. But then you say that "the owner doesnt care".
If you mean that the theft occurred took place in/at the store, then the victim is not necessarily the owner. It would well be a customer, an employee, or bystander. There the owner's preference as to court proceedings is irrelevant.
Is it necessary for the victim to care about being victim?
It is unclear what you mean. Even if the victim is not interested in charges being pressed, there are various reasons why the prosecutor might still opt to proceed. For instance, declining to prosecute the crime encourages the offender to further his illegal activity[-ies]. Likewise, the victim might be afraid that prosecution could unveil his illegal involvement in this or other matter. In that case, the prosecutor would be interested in moving forward with court proceedings.
Sometimes a victim's preference that no charges be pressed can dissuade the prosecutor to move forward. That happens when the crime is not serious and/or the victim's cooperation is crucial for a successful prosecution.