How purple do you want it?
Daylight on any planet is the result of a 3 step process:

1. The star gives off light.
2. The light changes as it travels through the planets atmosphere.
3. The light is perceived by a creature.

According to Hyperphysics, "Stars approximate blackbody radiators and their visible color depends upon the temperature of the radiator." This means there are very strict limits on what can happen in step 1. Also, I am assuming that there is nothing in step 3 you can change [i.e. the creatures on the planet are humans with normal color vision], because otherwise you enter complicated philosophical questions about what color even means.

In step 2 you, as the writer, have complete freedom. You can change what the atmosphere do whatever you want. The catch is that the less the planet is like Earth, the less likely your readers are to accept it. So we need to get a purple sky while minimizing the change in the atmosphere. That leaves the question: how purple do you want your sky?

Option 1: The first human to walk on the planet's surface says, "The sky is a little darker, just barely. Almost like a really, super light purple."

Option 2: The first human to walk on the planet's surface says, "Man, that sky is $^#&ing purple. I mean kids are going to be grabbing that purple crayon so much!"

Option 1: The super easy
The CIE diagram shows where colors are in CIE color space, and fortunately for us, there is a curve that goes through it that corresponds to every blackbody radiator called the Black Body Locus or Plankian Locus [image source]:

This curve does start to dip down toward the line of purples, which is that flat edge right on the bottom of the colored region. For step 1 simply pick a temperature for your star that corresponds to the color on this curve you want. Wikipedea currently has a GIF showing several examples. Then have the atmosphere do very little in step 2. A close up on the Black Body Locus is available here, and in it you can see that even Earth's atmosphere does not change the color of sunlight that much. The Daylight Locus depicts what a black body would look like after Earth's atmosphere changes it:

As long as you are careful with the habitability zone of your star and the UV radiation, you can get the desired result even with an Earth-like atmosphere!

Option 2: I have been working on this answer for 4 hours because of this option

We are now going to have to talk about why violet is cursed and evil and makes colometry an absolutely horrible field to study. To oversimplify, a spectral power distribution is a graph showing how much light of each wavelength is produced by a light source. You can go from an SPD to any color, but not the other way around. Here are 3 hypothetical SPD and my guess as to what color they produce [It's 12:40 AM, don't judge me]:

If the SPD is flat with one peak, then the color will be near the curve on the CIE diagram. This is where those charts showing what wavelength correspond to what colors come from. If the SPD is flat with one trough, then the color with be near the line of purples. If there is more than one peak or more than one trough, then there is no easy way to predict the resulting color and you need to do the calculations.

The fact that purples are caused by troughs is horrible news for us because black body radiators always look like hills, with a single peak and slopping sides. This means that your atmosphere is going to have to do a lot of work.

The star would need to produce a roughly flat SPD over the visible spectrum and the atmosphere needs to absorb a absolutely ton of light across a small band of frequencies. These do exist and are called absorption bands. However, every absorption band is associated with a specific molecule, so you can't just casually make them up. This would give you a SPD like my second example, but you would have to do a lot of research to find the right chemical and a lot of world-building based on what this chemical does in the atmosphere, or your readers may no accept it.

Unfortunately, we cannot take advantage of the fact that this is a binary star system to get an SPD that looks like my third example. The CIE diagram has the extremely useful property that if you pick any two points and find their midpoint, then mixing those two colors will result in the midpoint's color. This means the convex hull of the Black Body Locus is all the colors that could possibly result form combining starlight. As there are no colors there I would call purple, this super cool option that I was extremely pumped for would fail.