Comparison of the product structures in algebraic and in topological $K$-theory

Abstract

The compatibility up to sign of the product structures in algebraic K-theory and in topological K-theory of unital Banach algebras is established in total degree ≤ 2 . This answers a question posed by Milnor. 1 Statement of the theorem and de nition of the product structures in K-theories As an application of the computations made in [7], we prove the following result. 1.1 Theorem. Let A and B be two unital Banach algebras. Then the diagram K p (A)⊗K q (B) ? K p+q(A⊗Z B) Kp(A)⊗Kq(B) φp ⊗ φq ? ×Kp+q(A⊗B) (−1)φp+q ? commutes for p, q ≥ 0 satisfying p+q ≤ 2 . In other words, the external product structures in algebraic and in topological K-theory of unital Banach algebras are compatible in total degree ≤ 2 , up to the sign (−1) . In particular, for commutative unital Banach algebras, the internal product structures are also compatible in the same range and up to the same sign. Let us explain the notations. For a unital Banach algebras A (always over C), we denote by GL(A) the in nite matrix group with the usual direct limit topology, by E(A) the group of in nite elementary matrices, which coincides with the commutator subgroup [GL(A), GL(A)] of GL(A) , and by St(A) the in nite Steinberg group of A with standard generators (xij(a))i 6=j, a∈A . The algebraic and topological K-theory groups are de ned by : 1Research supported by Swiss National Fund for Scienti c Research grant no. 20-56816.99 2Research supported by Swiss National Fund for Scienti c Research grant no. 20-50575.97