On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutativeC*-algebras and present a proof in the finite dimensional case.

Groups with many characteristically simple subgroups

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

VI. On a collection of rock specimens from the Island of Socotra

The specimens of rocks—about 500 in number—collected by Professor Bayley Balfour during his late exploration of the Island of Socotra, were forwarded to me for examination. Several of these specimens, as was to be expected under the circumstances, were in a condition unfavourable for precise determination, being often fragments from weathered surfaces and sometimes much decomposed. Each, however, has been described as far as the circumstances would admit, and microscopic slides have been prepared for me by Mr. Cuttell from about 80 of the more interesting. As it happens, certain of these offer difficulties which in the present state of our knowledge are almost insuperable. While the use of the microscope has dispelled much confusion in our petrological ideas and supplied us in many respects with a firm basis of knowledge, it has not in every case—owing to the novelty of this mode of research and the inherent difficulties—enabled the student to feel perfect confidence in some of his conclusions, especially when he is restricted to this method of examination. Perhaps the greatest of our petrological difficulties is the distinguishing in every case between certain highly metamorphosed rocks and those of similar chemical composition which are truly igneous. It has, indeed, been maintained by some eminent geologists that certain sedimentary materials may be so altered by the combined action of water, heat, and pressure as to be converted in situ into a rock indistinguishable from one of those commonly held to be of igneous origin. Accordingly we read not seldom of “metamorphic granite” and of “gneiss passing into granite,” for it is of these that the above opinion is commonly held. Other geologists, indeed, go yet further and make a similar assertion, not only of the more coarsely crystalline rocks such as syenite, diorite, and gabbro, but even of the more compact varieties of felstone and greenstone, which in like manner are said to afford indubitable examples of transition into beds of true sedimentary origin. As regards this view, we may admit that if a mass of clastic materials be once reduced to a molten magma its past history is obliterated; and, further, that there is no reason, so far as we know, why this melting down should not occur. In this sense, any igneous rock whatever may possibly deserve the name of metamorphic. But, while admitting the à priori possibility of such a change, I must confess to being sceptical as to whether any case of it has yet been fully proved to exist. I have studied not a few of the alleged instances patiently and, as I trust, without prejudice, always with the following results—either that there was no evidence which was conclusive on either side, or that to a practised eye there was very clear evidence against the asserted transition— i. e ., that (to refer to the instance named above) either the so-called granite was clearly only a granitoid gneiss or else that it was distinctly intrusive in the schistose rock. In these investigations the microscope is a very great help, but I freely admit that there are many cases where we cannot rely upon it alone, and must also study the rock in the field. Our knowledge at present does not enable us to pronounce upon the classificatory value of certain structures which we observe in the microscopic study of some specimens. This difficulty, however, is one which time and experience will probably remove. To admit the existence of cases where it is at present safer to suspend the judgment in no way concedes that it is impossible ultimately to arrive at a conclusion. So then, while not professing in every case from examination of hand specimens alone to decide whether a rock is a granitoid gneiss or a true granite, I believe in the distinctness of the two rocks. As it happens, some of the specimens from Socotra belong to this dubious class; and I cannot say positively, even after microscopic examination, whether in certain cases we have a granitoid gneiss or a true granite, and in others a hornblendic rock of sedimentary origin or a true diorite.