It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking suspensions). I am wondering if other generalized cohomology theories have their own periodicity theorems, and if there is a general framework for conceptualizing them. I am interested in any substantive answer to this question, but there are two specific avenues for generalization that I am particularly curious about.
The first avenue begins with the Clifford algebra approach to Bott periodicity. This approach relates periodicity in K-theory to a certain natural periodicity present in the theory of complex Clifford algebras, and it generalizes the 8-fold periodicity of real K-theory (corresponding to an 8-fold periodicity in real Clifford algebras). Can one fruitfully generalize the notion of a Clifford algebra, associate to it a generalized cohomology theory, and analogously produce a periodicity theorem?
The second avenue involves Cuntz's proof of Bott periodicity for C*-algebras (which in particular implies topological Bott periodicity by specializing to commutative C*-algebras). Cuntz proves Bott periodicity for any functor from the category of C*-algebras to the category of Abelian groups which is stable (i.e. insensitive to tensoring with the C*-algebra of compact operators on Hilbert space), half exact, and homotopy invariant. The proof uses topological properties of Toeplitz algebras in an essential way. Because of the generality of his approach, I am left wondering if the essential features of his argument can be translated into more general contexts.
Any ideas are welcome!