Adjunction (single element)

The adjunct of a single element is used in mathematics when one wants to embed a ring without a single element in a ring with a single element, for example in order to be able to apply a theorem that only applies to rings with a single element.

Rings

Be any ring. Then define the operations on the Cartesian product ${\displaystyle A}$ ${\displaystyle A\times \mathbb {Z} }$

${\displaystyle (a,\lambda )+(b,\mu )\,=\,(a+b,\lambda +\mu )}$

${\displaystyle (a,\lambda )\cdot (b,\mu )\,=\,(ab+\lambda b+\mu a,\lambda \mu )}$,

whereby . Note that products can be formed using the obvious - module structure. Simple calculations show that with these operations there is a ring with the one element . If one identifies with, one can understand an element as writing and as a subring of . The above definitions are then written in the following expected form: ${\displaystyle a,b\in A;\,\lambda ,\mu \in \mathbb {Z} }$ ${\displaystyle \lambda b}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle A_{1}:=A\times \mathbb {Z} }$ ${\displaystyle e:=(0,1)}$ ${\displaystyle A}$ ${\displaystyle A\times \{0\}\subset A\times \mathbb {Z} ,}$ ${\displaystyle (a,\lambda )}$ ${\displaystyle a+\lambda e}$ ${\displaystyle A}$ ${\displaystyle A_{1}}$

${\displaystyle a+\lambda e\,+\,b+\mu e\,=\,a+b+(\lambda +\mu )e}$

${\displaystyle (a+\lambda e)\cdot (b+\mu e)\,=\,ab+\lambda b+\mu a+\lambda \mu e}$.

This means that each ring can be embedded in a ring with a single element. If you already had a one element, you get a new one, the original one of is no longer a one and the characteristic of is 0, even if it had a positive characteristic. ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A}$ ${\displaystyle A_{1},}$ ${\displaystyle A_{1}}$ ${\displaystyle A}$

In the above construction, a two-sided ideal is in and it applies . Since there is no zero divisor , even a prime ideal is in . ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{1}/A\cong \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle A}$ ${\displaystyle A_{1}}$

Algebras

If there is not only a ring but even an algebra over a solid , the above construction can be adapted so that the resulting ring is again an algebra. This one has merely by replacing, which means you then forms . The -algebra structure is given by the formula ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle K}$ ${\displaystyle A_{1}:=A\oplus K}$ ${\displaystyle K}$

${\displaystyle \mu \cdot (a+\lambda e):=\mu a+\mu \lambda e}$

given. When the adjunct of a unit is mentioned in the context of algebras, this construction is usually meant. Again, a two-sided ideal is in and it holds . Since there is a body, there is even a maximal ideal in . ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{1}/A\cong K}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle A_{1}}$

Normalized algebras

If a standardized algebra or even a Banach algebra is above , where for or stands, one can also make a standardized algebra in which one ${\displaystyle (A,\|\cdot \|)}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A_{1}}$ ${\displaystyle \mathbb {K} }$

${\displaystyle \|a+\lambda e\|:=\|a\|+|\lambda |}$

puts. That certainly makes a normalized space, and the multiplicative triangular inequality of carries over to , because ${\displaystyle A_{1}}$ ${\displaystyle (A,\|\cdot \|)}$ ${\displaystyle (A_{1},\|\cdot \|)}$

${\displaystyle \|(a+\lambda e)\cdot (b+\mu e)\|}$=  : = = = . ${\displaystyle \|ab+\lambda b+\mu a+\lambda \mu e\|}$ ${\displaystyle \|ab+\lambda b+\mu a\|+|\lambda \mu |\leq \|a\|\|b\|+|\lambda |\|b\|+|\mu |\|a\|+|\lambda ||\mu |}$ ${\displaystyle (\|a\|+|\lambda |)(\|b\|+|\mu |)}$ ${\displaystyle \|a+\lambda e\|\cdot \|b+\mu e\|}$

Is a Banach algebra, that is, as normed space completely , it is also a Banach algebra. ${\displaystyle A}$ ${\displaystyle A_{1}}$

If there is a -Banach algebra with involution , the involution can be given by the formula ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a\mapsto a^{*}}$

${\displaystyle (a+\lambda e)^{*}:=a^{*}+{\overline {\lambda }}e}$

to expand. If the involution is isometric , the same is true for . ${\displaystyle A_{1}}$ ${\displaystyle A}$ ${\displaystyle A_{1}}$

C * algebras

If a C * -algebra has no unit element, the above construction does not produce a C * -algebra . But you can have a different standard to choose the algebra also makes a C *. To do this, one sets ${\displaystyle A}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{1}}$

${\displaystyle \|a+\lambda e\|:=\sup\{\|ab+\lambda b\|;\,b\in A,\|b\|\leq 1\}}$.

This is precisely the operator norm for left-hand multiplication . ${\displaystyle L_{a+\lambda e}:A\rightarrow A,b\mapsto (a+\lambda e)b=ab+\lambda b}$

swell

• Jacques Dixmier : Les C * -algèbres et leurs représentations (Les grands classiques Gauthier-Villars). Éditions Gabay, Paris 1996, ISBN 2-87647-013-6 (unchanged reprint of the Paris 1969 edition)
• Louis H. Rowen: Ring Theory, Vol. 1 (Pure and applied mathematics; Vol. 127). Academic Press, Boston, Mass. 1988, ISBN 0-12-599841-4 .