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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $A$ be an associative algebra over $\\mathbb{K}$, with or without a unit.\nThe *unitization* of $A$ is the vector space $A^+ = \\mathbb{K} \\times A$ with the product\n$$(\\lambda, a)(\\mu, b) = (\\lambda \\mu,\\ \\lambda b + \\mu a + a b).$$\nIt is an associative algebra over $\\mathbb{K}$ with unit $(1, 0)$.\nThe map $a \\mapsto (0, a)$ embeds $A$ in $A^+$ as a [[two-sided ideal|0CD87SFM2N]] of codimension $1$.\n\n## Remarks\n\nOne writes $(\\lambda, a)$ as $\\lambda 1 + a$.\nEvery algebra homomorphism $\\varphi$ from $A$ to a unital algebra $B$ extends uniquely to a unital homomorphism $A^+ \\to B$, by $\\lambda 1 + a \\mapsto \\lambda 1_B + \\varphi(a)$.\nIf $A$ is a [[normed algebra|4DJ4V5VBSH]], then $\\norm{(\\lambda, a)} = \\abs{\\lambda} + \\norm{a}$ makes $A^+$ a normed algebra, complete when $A$ is complete.\nIndeed $\\abs{\\lambda \\mu} + \\norm{\\lambda b + \\mu a + a b} \\le (\\abs{\\lambda} + \\norm{a})(\\abs{\\mu} + \\norm{b})$.\nFor a C*-algebra $A$ one uses another norm on $A^+$, which makes it a C*-algebra.\nThe construction adds a new unit even when $A$ already has one.\n\nExample: for $A = C_0(\\R)$, the complex continuous functions that tend to $0$ at infinity, $A^+$ is isomorphic to the functions $f + \\lambda$ with $f \\in A$ and $\\lambda \\in \\C$.\nThese are the continuous functions on $\\R$ with one common limit at $+\\infty$ and $-\\infty$.\nThey are the continuous functions on the one-point compactification of $\\R$.\n\nNon-example: the componentwise product $(\\lambda, a)(\\mu, b) = (\\lambda \\mu, a b)$ on $\\mathbb{K} \\times A$ gives no unitization when $A \\neq 0$.\nFor it, $(1, 0)(\\mu, b) = (\\mu, 0)$, so $(1, 0)$ is not a unit.\n",
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