Grammar JLPT Level N3 - Lesson 6

では and じゃあ are both abbreviations of それでは, and used as conjunctions in Japanese. This structure may be considered as a combination of それ, the case marking particle で, and the binding (adverbial) particle は, but is commonly treated as a single word.
それでは has two primary functions. It can be used to continue in the same topic, showing that '(B) is a result of (A)'. Or it may be used to change the conversation topic completely, showing that '(A) is at a good point to become (B)'. In each of these cases, it can be translated as 'and with that' in English. 'If so', 'well then', and 'in that case' are also common translations.

• それでは、あなたは行(い)きたくないという事(こと)ですか？
  In that case, you mean that you don't want to go?
• では、こちらの商品(しょうひん)はどうでしょう？
  Well then, how about this product over here?
• 用意(ようい)できた？じゃあ、5分(ふん)で出(で)よう。
  Are you ready? In that case let's leave in 5 minutes.

The translation of 'in that case' is most commonly used when one person is making an assumption/drawing a conclusion about something that someone else has just said.

• それでは、あなたがやったってことでいいですね？
  In that case, would it be correct to say that you did it?