Unitization

Let A be a C^*-algebra having no unit. Then A^~=A direct sum C as a vector spaces together with

1. (a,lambda)+(b,mu)=(a+b,lambda+mu).

2. mu(a,lambda)=(mua,mulambda).

3. (a,lambda)(b,mu)=(ab+lambdab+mua,lambdamu).

4. (a,lambda)^*=(a^*,lambda^_).

5. ||(a,lambda)||=sup{||ab+lambda||:b in A,||b||<=1}.

is a C^*-algebra with the identity (0,1) and also a|->(a,0) is an isometrically *-isomorphism from A into A^~. The algebra A^~ is called the unitization of A.

For example, the minimal unitization of C^*-algebra C_ degrees(X) of continuous complex-valued functions on X vanishing at infinity is the C^*-algebra C(alphaX) of continuous complex-valued functions on the compact space alphaX, where alphaX is the one point compactification of X (Wegge-Olsen 1993).

This entry contributed by Mohammad Sal Moslehian

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Moslehian, Mohammad Sal. "Unitization." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Unitization.html

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