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"under its current name" is correct and usable in written English.
You can use this phrase when you are referring to something that has had its name changed at some point in time. For example: "The building was first opened in 1980 under its current name, King's Manor Museum."
Exact(29)
Cablevision acquired Fuse, then called MuchMusic USA, from a Canadian cable company, CHUM Television, in 2001 and introduced it under its current name in 2003.
It has been trading under its current name since 1806, it boasts 250 kinds of whisky, a very well-kept cellar and no jukebox (bliss).
Two years later the project was adopted by the SADC to which all five countries belonged but lack of progress led SADC tourism ministers to relaunch it in July 2003 under its current name.
In reality, roulette was derived in France in the early 18th century from the older games hoca and portique, and it is first mentioned under its current name in 1716 in Bordeaux.
On March 29 , 1938 it was incorporated as an independent company under its current name and sold for $3,500,000 to the former World War I ace Edward V. Rickenbacker and some associates.
It was first published in 1940 as a preface contributed by Tolkien to a translation of Old English poetry; it was first published as an essay under its current name in the 1983 collection The Monsters and the Critics, and Other Essays.
Similar(29)
John Edward Gray gave it its current name in 1846.
(It adopted its current name in 2000).
It assumed its current name in 1920.
It adopted its current name in 1946.
Under William, the castle acquired its current name of Sutton Valence.
More suggestions(15)
under its current pricing
under its current policy
under its current music
under its current slogan
under its current coach
under its own name
under its modern name
under its current setup
under its current structure
under its current Kabletown
under its current owner
under its previous name
under its current chairman
under its current governing
under its constitutional name
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com