Exponential sets and their geometric motions

Let us call an “exponential set” in a C*-algebraA any set consisting of the exponentialse ^X of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG ⁺ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G⁺. An exponential setC ? G⁺ inherits the geometric structure of the space G⁺ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G⁺ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G⁺. In particular we find thatC itself operates on C by the actionL _g a = (g^?1)*ag^? of the groupG of all invertible elements ofA in G⁺, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc ε C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace. The conditions onH are all additions to the following basic situation:H is the kernel of a (bounded linear) projection Φ:A → A. For example, ifH is closed under triple brackets X, Y, Z]] then parallel transport in G⁺ along geodesics inC through 1 ∈C preserves vectors tangent toC. Similarly, if the symmetric part of e ^X Ye ^?X ,Z] is inH for allX, Y, Z ∈H ^s thenC is “geodesically convex” in the sense that geodesics tangent toC stay inC. The most interesting cases correspond to a conditional expectation. Two additional conditions produce the groups described in the first paragraph: the case of a Z₂-graded C*-algebra with Φ the projection on the elements of degree 0 (which is automatically a conditional expectation) and the case of a conditional expectation such that the anti-symmetric part ofe ^X Ye ^?Y is in the range of Φ wheneverX, Y are self-adjoint and Φ(X)= Φ(Y) = 0. This is verified for example in the case of central traces.