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The density theorem for hermitian K-theory

Karoubi’s density theorem was first proved in Benayat’s thesis and then cited and used in several books and articles. As K-theory is a special case of hermitian εL-theory, a natural question is whether such a theorem is still true in the latter theory. The purpose of this article is to show that it is indeed the case. | Turk J Math (2018) 42: 2380 – 2388 © TÜBİTAK doi:10.3906/mat-1610-40 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article The density theorem for hermitian K-theory Mohamed Elamine TALBI∗, Department of Mathematics, University of Saad Dahleb, Blida, Algeria Received: 25.10.2016 • Accepted/Published Online: 03.07.2018 • Final Version: 27.09.2018 Abstract: Karoubi’s density theorem was first proved in Benayat’s thesis and then cited and used in several books and articles. As K-theory is a special case of hermitian ε L -theory, a natural question is whether such a theorem is still true in the latter theory. The purpose of this article is to show that it is indeed the case. Key words: Hermitian algebra, sesquilinear form, hyperbolic form, spectrum, polar form, K-theory 1. Introduction Functors of algebraic topology are numerous and often easy to define. However, although their algebraic properties are well known, the major task and the most difficult problem is to compute their values for interesting objects. Karoubi’s density theorem [2,5,10] says that if A is densely and continuously included in the Banach algebra B and units of A are those of B belonging to A, then K (A) and K (B) are isomorphic; it was first proved in Benayat’s thesis [2] to compute the K-theory of the Banach algebra of absolutely summable Laurent series in n variables. The theorem raises the question (known as Swan’s problem) whether, under the same hypotheses, there is equality of stable ranks [8]. It also allows the extension of topological K-theory to a whole class of dense subalgebras of the algebras involved and to Frechet algebras and dense subalgebras of these [1,6–8]. Since K-theory is a special case of hermitian ε L -theory, it is natural to ask whether the density theorem is still valid for the latter, allowing us to extend the mentioned problems to the hermitian situation; in this article, we answer the question positively. We refer to [4] for .