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Isometries of operator algebras. (English) Zbl 0045.06201
Let $$X$$ 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.
Reviewer: Edwin Hewitt

##### MSC:
 46L05 General theory of $$C^*$$-algebras
##### Keywords:
operator algebras; isometries
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