A. A. Rakhimov

In this paper the projectionless real $C^*$-algebras are investigated. Following construction of [4] a real $C^*$-algebra is constructed, which is separable, simple, nuclear, nonunital, and contains no nonzero projections. It is proved that a real $C^*$-algebra is projectionless if and only if the enveloping $C^*$-algebra is projectionless. An example of a projectionless real Banach ${}^*$-algebra with the $C^*$-property is constructed, the complexification of which contains a non-trivial projection.

В роботі досліджено безпроекційні дійсні $C^*$-алгебри. Використовуючи результати [4], побудовано дійсну $C^*$-алгебру, яка є сепарабельною, простою, ядерною, неунітальною, і яка не містить ненульових проекторів. Доведено, що дійсна $C^*$-алгебра є безпроекційною тоді і тільки тоді, коли огортуюча $С^*$-алгебра є безпроекційною. Побудовано приклад безпроекційної дійсної банахової ${}^*$-алгебру із властивістю $C^*$, комплексифікіція якої містить нетривіальний проектор.

It is known that a base for a traditional topology, or for a $L$-topology, $\tau$, is a subset ${\mathcal B}$ of $\tau$ with the property that every element $G\in \tau$ can be written as a union of elements of ${\mathcal B}$. In the classical case it is equivalent to say that $G\in \tau$ if and only if for any $x\in G$ we have $B\in {\mathcal B}$ satisfying $x\in B \subseteq G$. This latter property is taken as the foundation for a notion of strong base for a $L$-topology. Characteristic properties of a strong base are given and among other results it is shown that a strong base is a base, but not conversely.

We consider a non-commutative real analogue of Akcoglu and Sucheston's result about the mixing properties of positive L$^1$-contractions of the L$^1$-space associated with a measure space with probability measure. This result generalizes an analogous result obtained for the L$^1$-space associated with a finite (complex) W$^*$-algebras.