Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
What is von Neumann algebra?
algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator

Cycling is popular in Tucson due to its flat terrain and dry climate. Tucson and Pima County maintain an extensive network of marked bike routes, signal crossings, on-street bike lanes, mountain-biking trails, and dedicated shared-use paths. The Loop is a network of seven linear parks comprising over 100 mi (160 km) of paved, vehicle-free trails that encircles the majority of the city with links to Marana and Oro Valley. The Tucson-Pima County Bicycle Advisory Committee (TPCBAC) serves in an advisory capacity to local governments on issues relating to bicycle recreation, transportation, and safety. Tucson was awarded a gold rating for bicycle-friendliness by the League of American Bicyclists in 2006.
What kind of places to ride bikes does Tucson have?
marked bike routes, signal crossings, on-street bike lanes, mountain-biking trails, and dedicated shared-use paths

The first step toward Indian self-rule was the appointment of councillors to advise the British viceroy in 1861 and the first Indian was appointed in 1909. Provincial Councils with Indian members were also set up. The councillors' participation was subsequently widened into legislative councils. The British built a large British Indian Army, with the senior officers all British and many of the troops from small minority groups such as Gurkhas from Nepal and Sikhs. The civil service was increasingly filled with natives at the lower levels, with the British holding the more senior positions.
What appointments were  the first step in Indian self-rule?
councillors