### A note on states of von Neumann algebras

The author proves that on a von Neumann albebra (possibly of uncountable cardinality) there exists a family of states having mutually orthogonal supports (projections) converging to the identity operator.

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The author proves that on a von Neumann albebra (possibly of uncountable cardinality) there exists a family of states having mutually orthogonal supports (projections) converging to the identity operator.

Let L be a norm closed left ideal of a C*-algebra A. Then the left quotient A/L is a left A-module. In this paper, we shall implement Tomita’s idea about representing elements of A as left multiplications: ${\pi}_{p}\left(a\right)(b+L)=ab+L$. A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant ${\pi}_{p}{\left(A\right)}^{\text{'}\text{'}}$ of ${\pi}_{p}\left(A\right)$ in B(A/L) is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of A is carried out.