On the Universal C∗-algebra Generated by Partial Isometry

A universal C∗-algebra is constructed which is generated by a partial isometry. Using grading on this algebra we construct an analog of Cuntz algebras which gives a homotopical interpretation of KK-groups. It is proved that this algebra is homotopy equivalent up to stabilization by 2×2 matrices to M2(C). Therefore those algebras are KK-isomorphic. We recall the definition of partial isometry. Definition 1. Let H1 and H2 be Hilbert spaces and v : H1 −→ H2 be a bounded linear map. v is called a partial isometry if v∗v is a projection. We only want to emphasize the following important facts. The projection p = v∗v is called the support projection for v. Standard equations involving partial isometries are the following: v = vv∗v = vp = qv, v∗ = v∗vv∗ = pv∗ = v∗q. It is known that the above equations can be taken to define a partial isometry. First of all, our aim is to construct an involutive algebra generated by one symbol together with the above relations and prove the existence of a maximal C∗-norm on it. For the general construction and examples see [1]. Let U(v) be the universal involutive complex algebra generated by the symbols v, v∗. Let J(v) be the two-sided ∗-ideal generated by elements of the following form: (a) {v∗} − {v}∗, (b) {v} − {vv∗v}, where {v} and {v∗} denote the elements of U(v) which correspond to v and v∗ respectively. Then U(v) will denote the factor algebra U(v)/J(v), which is a complex ∗-algebra and has the universal property that if B is a 1991 Mathematics Subject Classification. 19K, 46L.