Is there a single word that means "just for show", meaning that someone is doing something outwardly to trick the world into thinking they are something that they are not?
"Ostensibly" is a good choice.
From Oxford Dictionaries online: "Apparently or purportedly, but perhaps not actually"
I believe there is no single word meaning "saying or doing something just for the show".
If you're looking for a noun defining the person who does it, I suggest
"a phoney" - an insincere or pretentious person (TFD)
"a beguiler" - someone who leads you to believe something that is not true (TFD)
"a pseudo (noun)" - a pseudo-scholar, a pseudo-intellectual. (a pseudo-intellectual would be someone who pretends to know more than he actually does and exhibits memorized information ostentatiously)
There is a term that comes close to this meaning:
If something is taken at face value then it is evaluated based on its outward appearance.
If you are looking for a more literal meaning:
A decortaion by its very definition is "just for show" and an ornament is specifically not supposed to be used, existing again "only for show"
If you mean the ideal of being showy but not having any depth, that is a person who care more for appearances than real things, then suitable words would be:
"Shallow", "Pretentious", "Fake" or "Image obsessed".
not candid or sincere
disingenuous could be used to describe someone is doing something outwardly to trick the world into thinking they are something that they are not.
existing or occurring at or on the surface
superficial as well can be used to describe someone is doing something outwardly to trick the world into thinking they are something that they are not.
A ruse, charade, subterfuge.
a trick or act that is used to fool someone, a wily subterfuge
His donation of millions to local charities was a ruse to mask his underhanded dealings with the scum of society.
an empty or deceptive act or pretense
Her attention to the duchess was a charade to garner the affections of the woman's wealthy husband for herself.
That person may be called a popinjay