Hilbert algebra

An algebra $A$ with involution (cf. Involution algebra) over the field of complex numbers, equipped with a non-degenerate scalar product $(|)$, for which the following axioms are satisfied: 1) $(x|y)=(y^{\ast }|x^{\ast })$ for all $x,y\in A$; 2) $(xy|z)=(y|x^{\ast }z)$ for all $x,y,z\in A$; 3) for all $x\in A$ the mapping $y\to xy$ of $A$ into $A$ is continuous; and 4) the set of elements of the form $xy$, $x,y\in A$, is everywhere dense in $A$. Examples of Hilbert algebras include the algebras $L_2(G)$ (with respect to convolution), where $G$ is a compact topological group, and the algebra of Hilbert–Schmidt operators (cf. Hilbert–Schmidt operator) on a given Hilbert space.

Let $A$ be a Hilbert algebra, let $H$ be the Hilbert space completion of $A$ and let $U_x$ and $V_x$ be the elements of the algebra of bounded linear operators on $H$ which are the continuous extensions of the multiplications from the left and from the right by $x$ in $A$. The mapping $x\to U_x$ (respectively, $x\to V_x$) is a non-degenerate representation of $A$ (respectively, of the opposite algebra), on $H$. The weak closure of the family of operators $U_x$ (respectively, $V_x$) is a von Neumann algebra in $H$; it is called the left (respectively, right) von Neumann algebra of the given Hilbert algebra $A$ and is denoted by $U(A)$ (respectively, $V(A)$); $U(A)$ and $V(A)$ are mutual commutators; they are semi-finite von Neumann algebras. Any Hilbert algebra unambiguously determines some specific normal semi-finite trace on the von Neumann algebra $U(A)$ (cf. Trace on a $C^{\ast }$-algebra). Conversely, if a von Neumann algebra $\mathfrak{A}$ and a specific semi-finite trace on $\mathfrak{A}$ are given, then it is possible to construct a Hilbert algebra such that the left von Neumann algebra of this Hilbert algebra is isomorphic to $\mathfrak{A}$ and the trace determined by the Hilbert algebra on $\mathfrak{A}$ coincides with the initial one [1]. Thus, a Hilbert algebra is a means of studying semi-finite von Neumann algebras and traces on them; a certain extension of the concept of a Hilbert algebra makes it possible to study by similar means von Neumann algebras that are not necessarily semi-finite [2].

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