The doctrine of Satan as understand by Christians today was not so developed in ancient Israel. At least, we have no indication in the Bible that such a doctrine was developed that shows Satan to be a spiritual archenemy of God. From Genesis to Revelation, there simply is no spiritual archenemy of God. As far as Scripture is concerned, satan simply means adversary or enemy. Jesus called Peter his enemy or his satan (G4567 – satanas) in Matthew 16:23. In the Old Covenant the Lord raised up two satans or adversaries (H7854) against Solomon (1Kings 11:14, 23). In fact, the Angel of the Lord was Balaam’s satan or adversary (H7854) in Numbers 22:22. So, what is Jesus referring to when he mentions the synagogue of satan?
When Jesus speaks of the “synagogue of Satan,” he is referring to the Jews, the physical descendants of Abraham. He is not saying they are the servants of an evil spirit being, who is the archenemy of God, but he is saying they are of the synagogue of the enemy of the people of God—i.e. enemies of the disciples of Jesus, who in this case reside in the church at Philadelphia (Revelation 3:9).
It was the Jews, the physical descendants of Abraham, who opposed the Gospel in the first century AD, and they were claiming to be the real Jews (i.e. the real people of God). Nevertheless, their claim was false, because they opposed the work of God in their generation. A people cannot be the people of God and oppose God at the same time. They slew the Messiah and persecuted his disciples, as they preached the good news (the Gospel) of Jesus’ coming reign and offer of eternal life to all who would submit to him.
Nevertheless, why would Jesus cause Jews to worship at the feet of the church at Philadelphia? Jesus didn’t mean Jews would suddenly fall down and worship mere men. The word used for worship (G4352 – proskuneo) is used for the worship of God, but it is also used to show respect for men of high rank. The same is true for its Hebrew counterpart (H7812 – shachah). It is used for worshiping God (Genesis 22:5) and also for expressing respect for men (Genesis 23:7). It describes Jacob bowing to his brother Esau seven times, as he approached him (Genesis 33:3). It also describes King Saul bowing himself before Samuel the prophet (1Samuel 28:14). Therefore, it is in this sense that Jesus said he would cause those (Jews) of the synagogue (assembly) of the enemy to come and worship, i.e. express their respect to the disciples of Jesus (Revelation 3:9).
In the early second century AD, Ignatius wrote a letter to the church at Philadelphia saying: “But if any one propound Judaism unto you, hear him not: for it is better to hear Christianity from a man who is circumcised than Judaism from one uncircumcised.” Although I don’t believe everything Ignatius wrote in his letters should be followed, this saying may have a lot to do with a great number of Jewish conversions after the Jew’s war with Rome in 70 AD. Many of them and later their descendants, understood the prophecy of the disciples of Jesus had come true in the destruction of the Temple (Matthew 26:64; cf. Acts 6:14). They, therefore, believed the Gospel, bowing to the disciples out of respect by admitting Jesus was the Messiah. Therefore, in them and also in gentiles who would also believe was fulfilled the prophecies of Isaiah (cf. Isaiah 45:14; 49:23; 60:14).
John seems to have interpreted Old Covenant prophecies that concern Israel in a manner that develops them in a spiritually restored Israel, i.e. Judah, plus the scattered ten tribes are united under Christ and given a new name. In addition to the uniting of the House of Judah and the House of Israel, the gentiles were also brought in under the same canopy of Christ as equal partners in the inheritance. The geo-political, restored Israel is rejected in favor of a people of righteousness, embracing all nations, which includes the Jews. This is the true Israel, restored as promised in the prophecies. Thus, those prophecies are not to be interpreted literally, but spiritually (Ephesians 2:11-22; 3:1-11). They don’t point to a literal, geo-political Israel, but to an Israel that comes not with observation (Luke 17:20-21).
In other words, one cannot point to this Israel on a map. So, when Jesus says “that they may know I love you” (Revelation 3:9), he is pointing to the believers’ vindication. That is, Jesus points to the time when what the disciples had been prophesying had come true (viz. the Jews’ war with Rome) to the astonishment of their persecutors and mockers. Then, after the dust of 70 AD settled, both Jew and gentile would know who had been telling the truth, and Jesus returned and judged Jerusalem and the Temple just as his disciples claimed he would do (Matthew 26:64; Isaiah 43:4, 9).
 See Ignatius, Letter to the Church of Philadelphia; Chapter 6. Ignatius is using the words circumcised and uncircumcised in their spiritual meaning. Anyone preaching the Law to Christians would be uncircumcised in heart, while anyone preaching Christ needs to be circumcised in his heart.