## Remarks on Professor Ola Bratteli

Ola Bratteli died, after years of ill health, on 8 February 2015 and later that year, on 11 June, a memorial meeting was held at which Erling Størmer gave the talk

*Remarks on Professor Ola Bratteli*. We give a version of Størmer's talk below:**Remarks on Professor Ola Bratteli; held at the meeting on 11 June 2015**

Ola Bratteli passed away on Sunday, 8 February 2015, at the age of 68 years, after several years of increasingly poor health. It was a very sad experience to see how he became weaker every time I saw him in recent years. His strength began to fail quite early. His last research article was published in 2008, and after that there was little he was able to contribute. So it was a premature end to his great career as a mathematician.

Ola was born on 24 October 1946. As a son of Trygve and Randi Bratteli he grew up in an active political environment, but he was never politically active. He was always supposed to be a mathematician. He studied arts subjects at the Oslo Cathedral School and then studied mathematics at the University of Oslo. I met him for the first time when he was about to begin his postgraduate thesis. Then he was a healthy looking dark-haired boy with a beard who often went on very long skiing trips. It soon struck me that he was a very efficient person who quickly developed a new theory. He had learned a lot about the field of operator algebras, in which he wanted to work. This branch of mathematics began in the years around 1930 when one wanted to develop a mathematical theory for quantum mechanics. The theory was further developed by mathematicians including John von Neumann, but there were few active participants until the late 1960s. Then the physicists arrived, and operator algebra became a popular field. So Ola came in at the right time when he graduated in 1971 with top marks. When it became clear to me how good his main work was, I told Ola that we had made a big mistake; this work should have been for a doctorate.

The work, published as the paper,

*Inductive limits of finite dimensional C*-algebras*(1972), made Ola well-known as a mathematician early in his career. Since it also provides a good example of how a mathematical theory develops, I will say a few words about it. The basic elements of the work are matrix algebra. If

*n*is an integer, then an

*n × n*matrix is a square consisting of

*n*rows that are written under each other each consisting of

*n*numbers. If

*m*is less than

*n*, then an

*m*×

*m*matrix can be included in an

*n*×

*n*matrix by placing it in the upper left corner of the

*n*×

*n*matrix. By placing several blocks of squares down the diagonal from the top left corner, one can describe all final dimensional operator algebras. The inclusion of such a matrix algebra in another can be described by a diagram that explains how the minor blocks are included in the major blocks.

When I was a student at Columbia University in the years around 1960, a student, James Glimm, who was a couple of years ahead of me, wrote a PhD thesis under the guidance of our joint supervisor Richard V Kadison, on operator algebra that was achieved by taking an infinite union of an increasing family of

*n*×

*n*arrays. This became a very well-known work, and when Ola was writing his postgraduate thesis, I suggested that he might be able to extend Glimm's results to some more general operator algebras, closely related to physics and similarly constructed. Ola quickly found out that he could generalize Glimm's work by studying infinitely increasing unions of matrix algebras. He then produced an infinitely large diagram describing all the inclusions and thus describing how the achieved operator algebra was constructed. His main result was that this diagram described the operator algebra completely, so that one obtained a classification of all such operator algebras, now called AF-algebras, an abbreviation for "approximately finite dimensional operator algebra". This result turned out to be far more important than Ola and I had foreseen. Analogue diagrams could be used in other areas of mathematics and with very good results. They are now called Bratteli diagrams and have become a familiar term among mathematicians.

After graduation, Ola studied for two years at New York University, where Glimm was, and it was a very good environment in mathematical physics. In addition to his studies, Ola spent a lot of time in the cultural life in New York and eating well. During the two years his appearance changed a lot; a lot of the hair was gone, the beard disappeared, and he put on so much weight that when I met him a few years later, I didn't recognize him, so I introduced myself to him.

Ola did not finish his Ph.D. at New York University, but went home and took his doctorate in Oslo in 1974. After that he went abroad. At that time, there was a very active and good environment in Marseille led by physicists who had developed the theory of quantum physics within operator algebra. Ola spent a lot of time there and became a very good friend and collaborator with an English colleague, Derek W Robinson. The two soon published several works together, and they continued to do so for the rest of Ola's career. Ola's two last research papers in 2008 include Robinson as co-author. They also wrote a two-volume work on operator algebra, see Ola Bratteli and Derek W Robinson,

*Operator algebras and quantum statistical mechanics*Vol. 1.

*C*- and W*-algebras, symmetry groups, decomposition of states*(Springer-Verlag, 1979); Ola Bratteli and Derek W Robinson,

*Operator algebras and quantum statistical mechanics*Vol. 2.

*Equilibrium states. Models in quantum statistical mechanics*(Springer-Verlag, 1981). It was specifically aimed at physicists and became very popular; it has become a standard reference for physicists working with operator algebras.

After a few years Robinson moved to Australia, so Ola travelled there several times. On one of the trips he stopped in Thailand where he was fortunate to meet Wasana, whom he married and with whom he had a good life.

But it was not just Robinson who was close to and a collaborator with Ola. In addition, there were four others he worked with a lot and who became his close friends. There was George Elliott in Canada, David Evans in Wales, Palle Jørgensen in the United States, and Akitaki Kishimoto in Japan. His mathematics was always about operator algebras and their relationship to quantum physics; in particular, the study of smooth derivations was central to Ola's research, see Ola Bratteli,

*Derivations, dissipations and group actions on C*-algebras*, Lecture Notes in Mathematics

**1229**(Springer-Verlag, Berlin, 1986).

It is interesting to look at Ola's bibliography of over 100 research articles. In the years after 1980, he published 84 works, of which only four have him as the sole author, but there are as many as 74 where he is one of at least five. I have mentioned some of his co-authors. There were often several other co-authors too. Ola was, on the whole, a very popular person to work with. He was, among other things, as I mentioned, very effective when he worked. Everyone liked Ola. He didn't say much when in a groups of several people, but on a two-person basis he spoke. Ola radiated something good, and you easily loved him. He was what I would call a good-hearted person. Our colleague Sergio Doplicher in Rome expressed this so well when he wrote to me, the same day that Ola died, after David Evans sent the news out to all his email addresses:

"We will greatly miss Ola, as a scientist and as a friend, his kind and soft approach to others, always based on understatement and totally antipodal to any form of arrogance, but with the constant dressing of his subtle humour, will stay with us for the rest of our lives."

Ola was a professor at Norwegian Institute of Technology (NTH) in the period 1980-1991, and was head of department in the period 1981-1984. From 1991 he was a professor at the University of Oslo. As a lecturer, he might have looked best at teaching at the postgraduate level. His style looked good, and he gave fine lectures that were very clear and well prepared. He was the Norwegian editor of the joint Scandinavian mathematics journal

*Mathematica Scandinavica*in the years 1987-1997.

He also received several awards for his research, among others were:

The Professor Ingerid Dal and Sister Ulrikke Greve Dal's scholarship in support of humanistic research in 2001.

The Nansen Foundation's prize for outstanding research from The Norwegian Academy of Sciences in 2004.

The Research Council of Norway's prize, the Möbius Prize, for outstanding research in 2004.