I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense that way.

The standard norm induced by the inner product on $\mathbb{C}^n$ is the Euclidean norm $ \sqrt{\langle x, x\rangle} = \| x \|_2 = \sqrt{\sum_i |x_i|^2}$. Similarly, endowed with the inner product $\langle A, B\rangle = \text{trace}(A^* B)$, the space of $n \times n$ complex matrices forms an inner product space with the induced Frobenius norm $ \|A\|_2 = \sqrt{\text{trace}(A^* A)}$.

However, there is a different norm that frequently comes up: it is the trace norm $\|A\|_1 = \text{trace}\left(\sqrt{A^* A}\right)$. There is a sense in which this norm is induced by the inner product on $\mathbb{C}^n$, since if $A = xx^*$, then $\|A\|_1 = \|x\|_2^2$.

However, what is the "meaning" of the trace norm? The Euclidean and Frobenius norms have an intuitive meaning in a geometric sense, as the length of a vector. Why do we care about the trace norm? Is it precisely because it is induced by the "natural" Euclidean norm on $\mathbb{C}^n$?

Additionally, if we express the trace norm in terms of the singular values of $A$, it corresponds to the L1 norm (i.e. sum of absolute values) of the singular values. Does the L1 norm on $\mathbb{C}^n$, i.e. $\|x\|_1 = \sum_i |x_i|$, have any interpretation, and does it share any similarity with the trace norm on $\mathbb{C}^{n\times n}$?

(I apologise if these questions are basic, I have asked this question on Math StackExchange and received no responses, and I could not find any information about the intuition for these norms and their interrelations.)