How do I in general realize a universal C\*-algebra generated by some generators and relation as concrete C\*-algebras? For example, I know that universal C\*-algebra generated by a single unitary is $C(\mathbb{T})$ by functional calculas. I am looking at the following examples to work on:


1. universal C\*-algebra generated by single self-adjoint element whose norm is 1.
2. universal C\*-algebra generated by single positive element whose norm is 1.
3. universal C\*-algebra generated by single normal element whose norm is 1.
4. universal C\*-algebra generated by single projection.



Realizing universal $C^*$-algebras as concrete $C^*$-algebras
**Edit**: as pointed out in the comments, the following answers the question for *unital* C\*-algebras presented in terms of generators and relations. When I say C\*-algebra, I really mean unital C\*-algebra.


It may depend on what exactly you mean by "concrete", but I highly doubt that there is a general solution to this; finding a concrete realization of a universal C\*-algebra requires classifying all the representations of the given generators and relations on Hilbert space, and this is an extremely difficult problem in general. For a more rigorous argument, see below.


But in your four examples, the universal C\*-algebras are all commutative, and simple answers are possible:


1. $C([-1,1])$.
2. $C([0,1])$.
3. $C(\mathbb{D})$ where $\mathbb{D}$ is the unit disk.
4. $C(\{0,1\})=\mathbb{C}^2$.


In each case, the generator is the identity function, just as in your $C(\mathbb{T})$ example. It is a good exercise to verify the required universal property in each of these cases.


Another good example is the C\*-algebra freely generated by two projections. This turns out to be the group C\*-algebra $C^\*(\mathbb{Z}\_2\ast \mathbb{Z}\_2)=C^\*(\mathbb{Z}\rtimes\mathbb{Z}\_2)$ and can be realized concretely as the subalgebra of $C([0,1],M\_2(\mathbb{C}))$ containing those matrix-valued functions which are diagonal on the endpoints $0$ and $1$. See [this paper](http://www.mscand.dk/article.php?id=1231) of Raeburn and Sinclair.


So why do I think that a general solution is impossible? Consider the word problem for groups: [there are](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.ijm/1256044631&page=record) groups given in terms of generators and relations for which there is no algorithm that can decide whether a given word in the generators represents the unit element. Now we can look at the maximal group C\*-algebra of such a group. This C\*-algebra is itself given by the same generators and relations together with additional relations requiring the generators to be unitary. If your intended meaning of a "concrete representation" comprises the existence of an algorithm that decides whether a given formal combination of generators represents $0$, then it follows that such a concrete representation cannot exist.


