Showing papers in "Bulletin of The London Mathematical Society in 1969"

On Extending Commutative Semigroups of Isometries

Abstract: Let V be an isometric operator defined on the Hilbert space «#, that is, ||Kx|| = Hxll for x in «5f. From a result due to von Neumann [5] and Wold [7], it follows that there is a unitary operator W defined on a Hilbert space Jf containing #? that extends V. An analogous result was obtained by Cooper [2] for a continuous one parameter semi-group of isometries. Independently, Ito [4] and Brehmer [1] showed that every commutative semigroup of isometries on Hilbert space can be extended to a corresponding commutative semigroup of unitary operators on a larger Hilbert space. It is the purpose of this note to give a more direct and natural proof of this latter result which is valid for Banach spaces and to prove certain ancillary results concerning the commutant of the semigroup of isometries. The proof is based on the construction of the direct limit of Banach spaces. A precise statement of the result will be given after this construction has been carried out. Let £ be a commutative semigroup and 3C be a Banach space. An isometric representation of £ on SC is a map o -> Va such that Va is an isometry on SC for each a in £ and Va VT = Vax for a and x in £.f Our object is to construct a Banach space <& containing SC and a representation a -> Wa consisting of invertible isometric operators on Va. We begin by constructing <&. A commutative semigroup possesses a natural order making it into a directed set, namely a > T if a = xy for some y in £. Let

On Homotopy Tori II

Abstract: In the preprint [2] of Kirby and Siebenmann, it is shown that vanishes for / # 3 and has order at most 2 in that case. It is further shown that isotopy classes of PL structures on a topological manifold of dimension ^ 5 correspond bijectively to reductions from Top to PL of the structure group of its stable tangent bundle. The same authors, using the same techniques, have since shown that 7r3(Top/PL) = Z2. We first use this to make some elementary observations on the homotopy type of certain spaces related to Top, and then apply this to study topological manifolds homotopy equivalent to the torus T" (n ^ 5): Our main result states that they are all homeomorphic to T". The argument showing that n3(Top/PL) = Z2 shows also that rc4(G/Top) is infinite cyclic, and contains n4(G/PL) as a subgroup of index 2. We take this as our starting point. It is well known that n3(PL) £ Z and iz4(G/PL) -> n3(PL) is an inclusion with index 24. The exact sequence

Free Associative Algebras

Abstract: The point of these notes is to recall some linear algebra that we’ll be using in many forms in 18.745. You can think of the notes as a makeup for the canceled class February 10. Vector spaces can be thought of as a very nice place to study addition. The notion of direct sum of vector spaces provides a place to add things even in different vector spaces, and turns out to be a very powerful tool for studying vector spaces. The theory of bases says that any vector space can be written as a direct sum of lines. In the same way, algebras are a very nice place to study multiplication. There is a notion analogous to direct sum, called tensor product, which makes it possible to multiply things even in different vector spaces. The theory of tensor products turns out to be a very powerful tool for studying algebras. I won’t write down the definition of tensor product (which you can find in lots of serious algebra books, or even on Wikipedia); but I will write the key property relating them to bilinear maps. Suppose U1, U2, and W are vector spaces over the same field k. A bilinear map from U1 × U2 to W is a function β : U1 × U2 →W (0.1)