Definition of the special unitary group

[Euge]

TL;DR Summary: Defining the special unitary group independent of matrix representation

In matrix representation, the special unitary group is distinguished from the more general unitary group by the sign of the matrix determinant. However, this presupposes that the special unitary group is formulated in matrix representation. For a unitary group action NOT formulated in matrix representation, what differentiates a special unitary group action from a more general unitary group action, such as in the infinite dimensional case?

If $H$ is a separable Hilbert space and $T$ is a trace-class operator on $H$ (i.e., a bounded linear operator on $H$ whose sequence of singular values is summable), then one may define the determinant of $I + T$ as the product $\prod_{j = 1}^\infty (1 + \lambda_j(T))$ where the $\lambda_j(T)$ are the eigenvalues of $T$. The space $B_1(H)$ of trace-class operators on $H$ is an ideal of the space $B(H)$ of bounded linear operators on $H$. So if $U(H)$ is the group of all unitary operators on $H$, it would make sense to define $SU(H)$ as the set of all $\Lambda \in U(H)$ such that $\Lambda - I \in B_1(H)$ and $\det \Lambda = 1$. This will be a subgroup of $U(H)$, and when $H = \mathbb{C}^n$, $U(H)$ and $SU(H)$ are canonically identified with the usual unitary matrix groups $U(n)$ and $SU(n)$, respectively.

 

[redtree]

$$GL(n,\mathbb{R}) = \{ T^{i j} \in V^i \otimes W^j, \dim{i}, \dim{j} = n: \det{T^{i j}} \neq 0 \}$$

$$SL(n,\mathbb{R}) = \{ T^{i j} \in V^i \otimes W^j, \dim{i}, \dim{j} = n: \det{T^{i j}} =1 \}$$

 

[mathwonk]

I am a bit puzzled by most of this discussion, since in finite dimensions neither the hermitian property nor the determinant one property are dependent on matrices. A hermitian linear map on a vector space V is one that preserves a hermitian inner product, and if V is finite dimensional, say dim(V) = n, then its nth wedge product V^n, is one dimensional, and thus the induced map on it is canonically given by a scalar, the determinant. The only issue for me is to define a determinant in infinite dimensions and Euge has addressed this. Of course even in finite dimensions, all unitary groups associated to vector spaces equipped with Hermitian forms, are isomorphic to the standard ones defined by matrices, so it is understandable to assume that matrices are natural to their definition.