Andrew M. Gleason

Measures on the Closed Subspaces of a Hilbert Space

In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function μ which assigns to every closed subspace a non-negative real number such that if {Ai} is a countable collection of mutually orthogonal subspaces having closed linear span B, then

Groups Without Small Subgroups

\mu (B) = \sum {\mu \left( {{A_i}} \right)}

The Definition of a Quadratic Form

.

Angle trisection, the heptagon, and the triskaidecagon

1. Introduction. The following elementary logical problem was a question in the William Lowell Putnam Mathematical Competition held in March 1953 (1): Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn, and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.

Remarks on the van der Waerden permanent conjecture

A topological group is said to have small subgroups if every neighborhood of the identity contains an entire subgroup of more than one element. It is easily deduced from the structure of the one-parameter subgroups that a Lie group does not have small subgroups. The possible existence of small subgroups in a locally euclidean group has long been recognized as one of the chief difficulties involved in Hilberts fifth problem. Our objective is to prove that every finite dimensional locally compact group which has no small subgroups is a Lie group. This partial solution of Hilberts fifth problem offers a new non-analytic criterion sufficent to insure that a group has an analytic structure. As such it represents a considerable step in the direction of ultimate solution of the problem. In fact, Montgomery and Zippin, using the present result inductively, have succeeded in proving the non-existence of small subgroups in locally connected groups of finite dimension. (See the following paper.) Together these results constitute an affirmative solution to Hilberts fifth problem. The proof depends on the construction of a suitable one-parameter semi-group of compact sets, a technique initiated by the author [3] to prove the existence of arcs. It seems unlikely that the strength of the method is now exhausted; on the contrary, there is definite hope of proving the further conjecture on the structure of locally compact groups: That every connected locally compact group is a projective limit of Lie groups.

How Does One Get So Much Information from So Few Assumptions

In order to apply our rather deep understanding of the structure of Lie groups to the study of transformation groups it is natural to try to single out a class of transformation groups which are in some sense naturally Lie groups. In this paper we iiltroduce such a class and commence their study. In Section 1 the inotioni of a l,ie transformation group is introduced. Roughly, these are grouips II of homeomorphismlls of a space X which admit a Lie group) topology which is stronlg enough to make the evaluation mapping (ht, x) ->h (x) of II X X into X continuous, yet weak enough so that H gets all the onie-parameter subgroups it deserves by virtue of the way it acts on X (see the definiition of admissibly weak below). Such a topology is uniquely determined if it exists and our efforts are in the main concerned with the questioni of wheni it exists anid how onie may effectively put ones hands on it wlheil it does. A niatural candidate for this so-called Lie topology is of course the compact-open topology for H. However, if one considers the example of a dense one-parameter subgroup H of the torus X acting on X by translation, it appears that this is not the general answer. In this example if we modifyv the compact-open topology by adding to the open sets all their arc componlents (getting in this way what we call the modified compact-open topology), we get the Lie topology of HI. That this is a fairly general fact is onie of our maini results (Theorem 5. 14). The latter theorem moreover shows that the reason that the compact-open topology was not good enough in the above example is connected with the fact that H was not closed in the group of all homeomorphisms of X, relative to the compact-open topology. Theorem 5. 14 also states that for a large class of interestinig cases the weakness condition for a Lie topology is redundant. The remainder of the paper is concerned with developing a certain criterion for deciding when a topological group is a Lie group and applying this criterion to derive a general necessary and sufficient conldition for groups of homeomorphisms of locally compact, locally connected finite dimensional metric spaces to be Lie transformiiation grouips. The criterion is remarkable in that local compactness is niot one of the assumptions. It states in fact

A note on locally compact groups

The study of quadratic forms is thereby reduced to the study of symmetric bilinear forms. There is something inappropriate about defining a quadratic form which is a function of one variable, in terms of a bilinear form which involves two variables. This raises the question of what requirements can be imposed on a function from V to F to define the set of all quadratic forms. The best known identity satisfied by quadratic forms is the parallelogram law

Projective topological spaces

Dr. Gleason graduated from Yale in 1942 and then served four years . . j in the Navy. After the war he went to Harvard as a Junior Fellow in the ;;, Society of Fellows. Except for an interlude in the Navy from 1950-52, he has been at Harvard ever since. He now holds the Hofis Professorship of Mathematicks and Natural Philosophy, a chair that was endowed in 1727. Although he has no doctors degree, he says that George Mackey was the equivalent of his dissertation supervisor. He has worked in several areas including topological groups, Banach algebras, finite geometries, and coding theory. He received the Newcomb Cleveland prize of the AAAS in 1952. He is a member of the National Academy of Sciences and is a former president of the American Mathematical Society.