Kapı açık olduğuna göre, arkadaşım eve gelmiş olmalı.

In Turkish, a present-tense sentence with a noun or adjective predicate often has no separate verb.

So:

• Kapı = the door
• açık = open

Together, Kapı açık simply means The door is open.

English needs is, but Turkish usually leaves that copula unspoken in the present tense.

Compare:

• Kapı açık. = The door is open.
• Kapı açıktı. = The door was open.
• Kapı açık olacak. = The door will be open.

So the lack of a word for is here is completely normal.

...olduğuna göre is a very common pattern meaning something like:

• given that ...
• since ...
• considering that ...
• if that is the case ...

So:

• Kapı açık olduğuna göre = Since the door is open / Given that the door is open

It introduces the reasoning basis for the conclusion in the second part of the sentence.

The speaker is thinking:

1. The door is open.
2. Therefore, my friend must have come home.

So this expression is often used for logical deduction.

It looks complicated, but it becomes clearer if you break it apart:

• ol- = to be / become
• -duğu = a suffix that turns a clause into something like the fact that ...
• -na = dative ending here
• göre = according to / considering

So olduğuna göre literally works like:

• according to the fact that it is ...
• more naturally: given that it is ...

Why use olmak here at all?

Because açık is an adjective, not a verb. Turkish often uses olmak inside these kinds of subordinate structures even when the main simple sentence would not need a verb:

• simple: Kapı açık.
• subordinate: kapı açık olduğu...

That is why you see olduğu even though the basic sentence has no separate is.

Because göre requires the dative case.

The word göre normally goes with -e / -a:

• bana göre = according to me
• sana göre = according to you
• duruma göre = according to the situation

So here:

• olduğu + na göre = olduğuna göre

If it were olduğunu, that would be a different case and would be used in different sentence patterns, for example after verbs like bilmek:

• Kapının açık olduğunu biliyorum. = I know that the door is open.

So olduğuna göre is correct because göre needs that dative ending.

Gelmiş olmalı expresses a deduction about a past action.

It means:

• must have come
• has probably come

Breakdown:

• gelmiş = came / has come, with an inferential or completed sense
• olmalı = must be / probably is

Together, gelmiş olmalı means the speaker is not directly witnessing the action, but is inferring it from evidence.

So the logic is:

• The door is open.
• Therefore, my friend must have come home.

This is one of the most natural Turkish ways to express must have done.

This is a very important difference.

• gelmeli = should come / must come
• gelmiş olmalı = must have come

So:

gelmeli

This expresses obligation, necessity, or expectation about the action itself.

• Arkadaşım eve gelmeli.
  = My friend should come home. / My friend must come home.

gelmiş olmalı

This expresses a conclusion about the past.

• Arkadaşım eve gelmiş olmalı.
  = My friend must have come home.

So -malı alone is often about what ought to happen, while -miş olmalı is often about what the speaker concludes has already happened.

Because gelmek uses the dative case for the destination.

• ev = house / home
• eve = to the house / home
• evde = in the house / at home

So:

• eve gelmek = to come home / come to the house
• evde would describe location, not destination

Compare:

• Arkadaşım eve geldi. = My friend came home.
• Arkadaşım evde. = My friend is at home.

English says come home, but Turkish marks the destination with -e.

Arkadaşım is:

• arkadaş = friend
• -ım = my

So it literally means my friend.

The possessive suffix changes by vowel harmony:

• arkadaşım = my friend
• evim = my house
• okulum = my school

Also, arkadaş can be broader than English friend in some contexts. Depending on the situation, it may mean:

• friend
• buddy
• companion
• classmate
• colleague

But in this sentence, my friend is the most natural reading.

Yes, but the nuance changes.

• Kapı açıksa, arkadaşım eve gelmiş olmalı.
• Kapı açık olduğuna göre, arkadaşım eve gelmiş olmalı.

Both can lead to something like the same overall meaning, but açık olduğuna göre sounds more like:

• given that the door is open
• since the door is open
• judging from the fact that the door is open

It feels more like a reasoned conclusion from evidence.

By contrast, açıksa is closer to:

• if the door is open

That can sound more conditional and less firmly based on an observed fact.

So in this sentence, açık olduğuna göre is especially good because the speaker is making a deduction from something they can see.

The comma is helpful in writing, but it is not the main thing that creates the meaning.

• Kapı açık olduğuna göre, arkadaşım eve gelmiş olmalı.

The first part gives the evidence/reason, and the second part gives the conclusion.

This order is very natural in Turkish:

1. evidence
2. conclusion

You could sometimes change the order for emphasis, but this version is the most straightforward and natural.

So the comma mainly marks a pause after the introductory reasoning clause. It is common and useful, especially in writing.