No. Outbound email is always queued as a draft for your review — you approve, edit, or reject it on the Approvals page or right in the chat. This is enforced in the application's architecture, not just in the AI's instructions: the assistant has no tool that sends directly.
The one exception is deliberate: standing instructions you create yourself (for example, "forward receipts to my accountant"). Those run with the trust level you gave them, and every action they take is still recorded in your Approvals history.