Isometries of operator algebras.

*(English)*Zbl 0045.06201Let \(X\) and \(Y\) be compact Hausdorff spaces, and let \(C(X, K)\) resp. \(C(Y, K)\) denote the sets of continuous complex-valued functions on \(X\) resp. \(Y\). It is known that a linear isometry of \(C(X, K)\) onto \(C(Y, K)\) (under the uniform metric for both spaces) is actually an algebraic isomorphism followed by multiplication by a function in \(C(Y, K)\) which has absolute value \(1\) [M. H. Stone, Trans. Am. Math. Soc. 41, 375–481 (1937; Zbl 0017.13502)]. A non-commutative analogue of this theorem is given here, dealing with a certain class of not necessarily commutative Banach algebras (= normed rings). A \(C^*\)-algebra is a Banach algebra admitting an adjoint operation \(x\to x^*\) satisfying axioms 1’)–6’) of I. Gel’fand and M. Neĭmark [Mat. Sb., N. Ser. 12(54), 197–213 (1943; Zbl 0060.27006)]. Gel’fand and Neĭmark (loc. cit., Theorem 1) have shown that every \(C^*\)-algebra is algebraically, normwise, and adjoint-preserving isomorphic to a uniformly closed algebra of bounded operators on some Hilbert space. Thus the author can, in discussing \(C^*\)-algebras, consider only algebras of operators.

Two preliminary results are obtained first: the extreme points of the unit sphere in a \(C^*\)-algebra \(\mathfrak A\) are the set of partially isometric operators \(U\in \mathfrak A\) where \(U^* U = E\), \(U U^* = F\), and \((I - F)\mathfrak A (1 - E) = 0\); the positive part of the unit sphere of \(\mathfrak A\) has as extreme points the projections in \(\mathfrak A\).

Turning next to isometries of \(C^*\)-algebras, the author proves: an isomorphism of a \(C^*\)-algebra \(\mathfrak A\) which preserves the *-operation is isometric and preserves commutativity; an isometric linear mapping \(\rho\) of a \(C^*\)-algebra \(\mathfrak A\) onto a \(C^*\)-algebra \(\mathfrak A'\) is a \(C^*\)-isomorphism followed by left multiplication by the unitary operator \(\rho(I)\).

The paper concludes with a classification of extreme points for factors.

Two preliminary results are obtained first: the extreme points of the unit sphere in a \(C^*\)-algebra \(\mathfrak A\) are the set of partially isometric operators \(U\in \mathfrak A\) where \(U^* U = E\), \(U U^* = F\), and \((I - F)\mathfrak A (1 - E) = 0\); the positive part of the unit sphere of \(\mathfrak A\) has as extreme points the projections in \(\mathfrak A\).

Turning next to isometries of \(C^*\)-algebras, the author proves: an isomorphism of a \(C^*\)-algebra \(\mathfrak A\) which preserves the *-operation is isometric and preserves commutativity; an isometric linear mapping \(\rho\) of a \(C^*\)-algebra \(\mathfrak A\) onto a \(C^*\)-algebra \(\mathfrak A'\) is a \(C^*\)-isomorphism followed by left multiplication by the unitary operator \(\rho(I)\).

The paper concludes with a classification of extreme points for factors.

Reviewer: Edwin Hewitt

##### MSC:

46L05 | General theory of \(C^*\)-algebras |