It might be possible for stars that emit more violet light than the Sun to have habitable planets, where the daytime sky might be more violet colored than blue.
The star of a habitable palnet might have a ring of dust particles that absorb ultraviolet light from the star and re emit it as violet light, and that might possibly color the day time sky more violet than Earth's sky.
A habitable planet in another star system might possibly have a ring of dust, and that dust might poossibly be heated by the star and emit violet light, which might possibly color the daytime sky of the planet more violet than Earth's sky.
A habitable planet should have some dust in its atmosphere, and if that dust is purple colored it might refelct enough light to possibly make the day sky appear more purple than Earth's does.
The star and planet might be passing through a nebula, and that nebula could reflect starlight from the star onto the planet. But It seems improbable that the nebula could be bright enough to change the color of the night sky, let alone the day sky.
A planet habitable for some lifeforms, possibly even habitable for lifeforms with similar requirement to humans, could possibly have a thinner atmosphere, which like the atmosphere of Earth at hight altitudes, was a much deeper blue, perhaps a purplish blue color.
Part One of Seven: A Star Emitting More Violet LIght.
Adrian Colomitchi suggested that a violet sky could come from having a star which emitted much more violet light than the Sun does, thus causing violet light to be scattered in every direction by air particles andmaking the skay appear violetinstead of blue.
Since it is a shorter wave length than blue, it will be Rayleigh-scattered stronger than blue. So your problem is to make the violet intensity in the light spectrum of your star higher than the intensity of the blue. Which is quite simple to do: just move the temperature of the star by 700-1000K above the Sun's one.
The Sun is a G2V type star with a surface temperature of 5,772 degrees Kelvin. So making the star 700 to 1,000 degreees K hotter than the Sun would give it a surface temperature in the range of 6,472 to 6,772 degreees K.
Spectral class F6V stars have 1.16 times the mass of the Sun and a surface temperature about 6,400 K, while spectral class F5V stars have 1.20 times the mass of the Sun and a surface temperature of about 6,545 K. Spectral class F4V stars have 1.23 times the mass of the Sun and a surface temperature about 6,690 K, while spectral class F2V stars have 1.31 times the mass of the Sun and a surface temperature of about 7,040 K.
So according to Adrian Colomitchi's suggestion a star in that range should have the right surface temperature to emitt more violet light than blue light, and thus make the skies of the planet appear violet instead of blue.
The Earth is about 4.6 billion years ago, and didn't have an oxygen rich atmosphere that humans and life formes with similar requirements could breath until about 500 to 600 million years ago, and thus 4 billion years after the planet formed.
If the story requires the planet to be habitable for humans or lifeforms with similar requirements, the planet will have taken billions of years to tobecomehabitable for them. And the star willhave had to stay on the main sequence with fairly steady luminosity for those billions of years. Thus the star will have to have a spectral type which can stay on the main sequence for enough billions of yeears.
[Unless, of course, in the story an advanced civilization terraformed the planet some time ago, giving it a breathable atmosphere before the planet wuld have developed one naturally}
Someone objected that a star hot enough to emit so much violet light would not last long enough on the main sequence to become habitable, and Adrian Colomitchi said that a star with 1.2 times the mass of the Sun (an F5V according to the table) would last for 6.34 billion years, which would be long enough.
The only scienctific study about planetary habitability for humans (and bengs with the same requirements) in particular, instead of liquid water using life in general, is Habitable Planets for Man, Stephen H. Dole, 1964.
On page 68 Dole says:
The only stars that conform with the requirement of stability for at least 3 billion years are main sequence stars hving a mass less than about 1.4 solar masses--spectral types F2 and smaller--although the relationship between mass and time of residence on the main sequence is probablyy not known with great accuracy and is subject to futue revision (see figure 25).
I note that Dole says that a star with a lifetime onthe main sequence of 3 billion years would have 1.4 times the mass of the Sun and would be an F2V type star. But the table in Wikipedia (which doesn't give the stellar lifetimes) lists a star with 1.4 times the mass of the Sun as an F0V, and a star with 1.31 times the mass of the Sun is listed as an F2V.
Any writer who wants to make the star of a human habitable world as hot and luminous as possible should investigate that descrepancy, and look up the latest information on how long stars of different spectral classes remain on the main sequence.
So it seems possible that a main sequence spectral class star less massive than F0or F2 might remain on the main sequence for at least 3 billion years and thus possibly have a planet which has already became habitable for humans (or being swiht similar environmental requirements).
But some scientists don't think that it is possible for spectral class F stars to have habitable planets. They think that their increased ultraviolet light could prevent life for developing or kill off life osono afterit develops. And they think that because a F class star spends less time on the main sequence it will grow hotter faster than a class G star, and thus that its circumstellar habitable zone will migrate outwards from it faster, which could mean that planets would not spend enough time in the habitable zones of class F stars.
Here are links to several discussions of the potential habitability of planets orbiting spectral class F stars.
So a writer who wants to give a habitable planet a purple or violet sky will have a problem with making the star hotter and emitting more violet radiation than blue radiation. Some more cautious writers might want to avoid putting habitable planets in orbit around and class F stars, and other might want to only use less massive and cooler class F Stars.
Part Two: A Star With a Ring of Dust.
I note that if the star in the system happens to be surrounded by rings of dust, that dust might absorb ultraviolet radiation and reemitt as violet radiation. That might possibly increase the amount of violet radiation the planet receives from its star.
Part Three: A Planet With a Ring of Dust.
And it is possible that the planet could have a ring of particles around it. And possibly the particles absorb ultraviolet radiation from the star and then emitt violaet radiation, which would increase the amount of violet radiation the planet receives.
Part Four: A Planet With Purple Dust in the Atmosphere.
And possibly the planet has dust in its atmosphere, as all Earthlike planets do, and possibly that dust is all violet colored and thus reflects violet light into the sky. And possibly that might increase the amount of violet light in the sky of the planet.
Part Five: A Planet Travelling Through a Nebula.
And possibly the star system happens to be travelling through a nebula.
And possibly that nebula looks a lot like a nebula in a science fiction movie or tv show had is thick and obapuge and reflects a lot of varicolored light. And maybe this nebula happens to be puble colored. So the sky of the planet might look purple at night, and manybe even in the day if the light reflected from the nebula is bright enough.
But movie and tv nebeulas are not realistic. They are based on astronomical photographs which show dense, opague, colar nebulas. And despite the old saying, those astronomical photographs lie. At least those photographs deceive people who don't realized that they are taken with hours long exposures though telescopes which are ketp constantly turning lsightly to keep the nepublas in the view.
The human eye usually processes 10 to 12 images per second. So each image you see has the brightness level of light which has accumulated for only 0.083333 to 0.1 secondexposure time.
There are 3,600 seconds during an hour. So a photograph exposed for an hour would recieve about 36,000 to 43,200 as much light on each picturee elelement as a human eye lookign at the same picture would receive while sensing one image. So an astronomical photo of a nebula exposed for several hours would receive over a hundred thousand times as much laight as you wuld see in each image looking at the nebula though the same telescope.
Nebulas seen by the human eye through a telescope are very delacate, almost transpartent, and pale. They look nothing like photos of nebulas exposed for hours at a time.
So if a fictional solar system is close to a nebula, the nebula would be visible at night as a pale and delicate wonder, but it probably wouldn't be bright enough to lightened the night sky from black to purple. And since daylight on the planet would be tens or hundreds of thousands of times as bright, the nebula light probably wouldn't be enough to make the color of the day sky change.
Part Six: A Planet With a Thinner Atmosphere Than Earth.
If you look at the sky on a clear day, you will see that it is lighter and paler near the horizon and gets brighter and bluer higher in the sky, until it is very blue at the zenith. That is because when you look toward the horizon you are looking through a greater distance of thick air, wich catters the sunlight more. But the air gets thinner with height. So as you look stright up your a looking through a thin layer of the thickest air, then a thin layer of slightly less thick air, and then a thin layer of even less thick air, and so on. The total amount of air you see scattering sunlight above you is less than when you look horizonatally toward the horizon.
Pictures taken at the peak of Mount Everest show a blue sky low down at the horizon and a darker and darker sky above the horizon. At the peak of Everest more than half o the atmosphere that scatters light is below, not above.
The peak of Mount Everest has an altitude of 8,848.86 meters, or 29,031.7 feet, above sea level. And the skylooks darker and more black from there than from lower altitudes. I guess some people might possibly say that the sky looks more purple from the peak of Everest.
The peaks of Mount Everest and some of the other tallest mountains in the world are in what mountaineers call "the death zone", above about 8,000 meters or 26,246.72 feet.
Most climbers in the death zone breath bottled oxygen, and most of them start using the bottled oxygen way below the death zone. And even using bottled oxygen, many climbers suffer from the thin air on high peaks.
People born at higher altitudes can go higher without bottled oxygen than people born at sea level. People from the high Tibetan plateau and the high Andes have the most tolerance for low oxygen levels. And people can train themselves to breath in thinner and thinner air in preparation for living and working in the higest towns and villages in the world, and for climbing tall mountains.
In fact, some mountaineers have performed the fantastic feet of climbing to the peak of Everest safely without bottled oxygen, and the even more fantastic feat of climbing back down the mountain safely afterwards without bottled oxygen despite their fatigue.
But people who try climbing Mount Everest without bottled oxygen have a survival rate which is much smaller than the survival rate of those who try to climb with bottled oxygen, and the survival rate of those who use bottled oxygen is not a very good one itself.
So that makes me think that if a group of people from Earth settled a planet with a slightly thinner atmosphere, their descendants would gradually adapt over generations to function as well there as we do on Earth. And if after many generations some of their descendants settled a planet with a even less dense atmosphere still, they would adapt after many generations to function just as well there as well we do on Earth.
And so, after settling many successive planets with succesivly slightly less dnsse atmospheres, and spending generations on each planet to adapat, a group of human descendants might possibly be able to settle a planet where the typical atmospheric density was similar to that at the peak of Everest, and where the sky was darker than on Earth, a dark blue or maybe a dark purplish blue color.
Insects can fly and kite at very high altitude. In 2008, a colony of bumble bees was discovered on Mount Everest at more than 5,600 metres (18,400 ft) above sea level, the highest known altitude for an insect. In subsequent tests some of the bees were still able to fly in a flight chamber which recreated the thinner air of 9,000 metres (30,000 ft).
Ballooning is a term used for the mechanical kiting that many spiders, especially small species such as Erigone atra, as well as certain mites and some caterpillars use to disperse through the air. Some spiders have been detected in atmospheric data balloons collecting air samples at slightly less than 5 km (16000 ft) above sea level. It is the most common way for spiders to pioneer isolated islands and mountaintops.
Some birds have been recorded to fly above 8,000 meters or 26,246.72 feet.
They include the alpine chough at 8,000 meters (26,500 feet) on Everest, whooper swans at 8,200 meters (27,000 feet) over northern Ireland, bar-headed goose at 8,800 meters (29,000 feet), the common crane at 10,000 meters (33,000 feet) over the Himalayas, and Ruppell's Vulture at 11,300 meters (37,100 feet).
So birds, multicelled animals who need oxygen to live, are able to fly at about the peak of Mount Everest, where the sky is much darker than at sea level. So alien lifeforms should be able to adapt to a planet where the atmosphere is a thin as on the summitt of Everest, and the sky is much darker than at sea livel on Earth.
Part Seven: Conclusion.
Obviously someone could try a combinatin of several of the suggested methods to make the daytime sky of the planet more violet or purple than that of Earth.
I note that such a planet could be approximately as habitable for Earth humans as Earth is, except for the method of the lanethaving a much thinner atmosphere than Earth. Such a world could be a lot different from worlds with purplish skies for other reasons.
And I could imagine a story where someone comes from a world with a purplish sky because its star emits more violet light than the Sun does, and they crash on some unexplored pplanet. They see thorugh the viewscreen or porthole that the planet has a purplish sky like their home world, and assume it has a similar atmosphere to home.
So they go out the airlock without testing the atmosphere and gasp for air because this is a world where the sky looks purplish because the air is much thinner than on their home planet.