الحمد لله رب العالمين، والسلاة والسلام على الرسول الكريم وعلى آله وصحبه أجمعين
This practice has been mentioned in some books of Shafi'e mazhab by later scholars of the mazhab. The authors themselves confirm that this practice has no proof from the Sunnah and athar including the rightly guided caliphs, but then mention the ruling of this practice as Bid'ah Hasanah! See: Haashiah al-Jamal 'ala Sharh Manhaj al-Tullab by Sulayman bin Umar al-Jamal (d. 1204h); and Haashiyah al-Bujairimi 'ala al-Minhaj by Sulayman bin Muhammad al-Bujairimi (d. 1221).
The correct view is the practice should be stopped because it is an innovation in the matter of the religion and the prophet said:
مَنْ عَمِلَ عَمَلًا لَيْسَ عَلَيْهِ أَمْرُنَا فَهُوَ رَدٌّ
"Whoever practises a deed that we have not commanded, then it is rejected." [Muslim]
And as Ibn Kathir mentioned in his Tafsir:
ولو كان خيرا لسبقونا إليه
"If it was good, surely (the companions of the prophet) would have beaten us to it."
Imam Al-Albaani (رحمه الله) mentioned in his listing of innovations done on Friday by muslims, no. 19 & 20 he mentioned:
19 - الترقية ، وهي تلاوة آية : إن الله وملائكته يصلون على النبي
20 -ثم حديث : (( إذا قلت لصاحبك ... )) يجهر بذلك المؤذنون عند خروج الخطيب حتى يصل إلى المنبر "
19. At-Tarqiyah - ie. the reciting of verse: "Verily Allah and the Angels send their prayers to the prophet ..." [33:56]
20. Then (mentioning) the hadith: "If you say to your friend, during Jumu'ah, Quiet!...", loudly by the muazzin while the Khatib comes out till he reaches the minbar."
Wal-hamdu lillahi rabbi l-'alamin.