The concept of authenticity in the classroom is an interesting one. It can be argued that as a teacher, the strength of your relationship with your students is of equal importance to the information you're teaching them. Without that established connection, it's easy for students to "check out", become distracted, or overall apply themselves less fully than they are capable of - no matter how well the teacher believes they are performing.
Presenting your authentic self in every day life can be challenging, but that challenge is only exacerbated for teachers responsible for managing a classroom and teaching effectively. Thinking back to my high school years, I can clearly recall teachers who tried hard to be "cool" in order to appeal to students...an effort that consistently backfired, and often caused students to lose respect for the teacher or take advantage of them. My favorite teachers, those I found to be most effective at their jobs as well as the most memorable, allowed students to see their personal side (discussing their family when appropriate, decorating their classroom with some personal effects, sharing interests, casual yet professional language) which helped students view them as real people and created a genuine bond between them. They created a balance between personal and professional, which suggested their authenticity, and allowed students to respond favorably and learn better.
Translating this approach to online learning can be difficult, especially in asynchronous learning. I do believe that face-to-face communication (even if those faces are presented virtually!) is a necessary component for establishing authenticity. Without at least video lessons or synchronous learning, it's very easy for teachers to be seen as a robot (considering Susan's fear from Cranton's paper) or as an ephemeral presence without much purpose in the classroom. As a teacher in online learning classrooms, I think the winning combination is clear, casual communication that also conveys the teacher's expert knowledge with firm, clear guidelines. This allows students to form a bond with the teacher, learn from them, and understand upfront what is expected of them.
Were I to lead an online classroom, I would design a synchronous program which encourages expert instruction, live student participation and community building. Similar to how we began our class at the beginning of this semester, I would dedicate the first class to introductions, providing time for both myself and all of my students to share a little and get to know each other. For following classes, I would start off by introducing the material, providing a synopsis of that week's concepts and any relevant context, as well as pose questions for the students to consider during class time. I'd encourage students to respond, ask questions, challenge me or the material, and discuss ideas. At the end, I'd leave them with some open ended questions for them to consider regarding how these concepts impact their life or their work. Of course, this is just a general framework - there would be lots of room to facilitate activities, utilize online learning resources, etc, but that would be my overall plan to establish my authenticity, connect with students, and allow them to participate and collaborate.