Youth hockey, adult hockey, and figure skating are underway at the Martha’s Vineyard Ice Arena while it awaits final inspection after a $3.7 million remodel, general manager Peter Lambos told The Times this week. The arena is closed to other activity, he said. Youth hockey hit the ice on Aug. 18 and played their first game on Aug. 24, he said. The Martha’s Vineyard Figure Skating Club also started up on Aug. 18. Two adult hockey leagues are slated to begin this week, he said. Mr. Lambos was hesitant to venture when the final inspection will take place because it has already been pushed forward several times. He said he was hopeful it would be soon, however.